Systems and methods for determining energy of prepared quantum states

Abstract

Provided are computer-implemented methods and quantum computing systems for preparing computational states representing quantum states of a physical system, including performing a computational evolution of the state and then determining physical properties of the system using the time-evolved computational state. A described computer-implemented method enables computing Hamiltonian dynamics of observables on a quantum computer to provide information about the physical system represented by the Hamiltonian. A described computer-implemented method prepares low energy electronic structure states of a physical system using adiabatic evolution. The method determines the energy of an equilibrium quantum state of a physical system using a time evolution operator that enables adiabatic evolution of a prepared electronic structure state on a quantum computing system. A described method involves indirectly determining the energy of an eigenstate of a physical system by evaluating the expectation values of a time-evolution operator averaged across multiple shots for randomly-generated quantum circuits. Example methods enable calculation of the energy of an evolved computational state with chemical accuracy, due to avoiding discretization errors, with a smaller circuit depth than known alternatives.

Description

[0001] Systems and methods for determining energy of prepared quantum states Field of Inven^on The present inven^on relates to quantum compu^ng and, in par^cular, to the prepara^on of low- energy electronic structure states and determina^on of the energy of the prepared states. The inven^on has applica^ons in computa^onal chemistry using quantum compu^ng systems, where the energy of a prepared state must be determined with high accuracy despite noise within the quantum compu^ng system. Background Quantum computers are inherently suitable for determining proper^es of complex physical systems, by modelling the energy states of individual molecules or solid materials, because they exploit quantum phenomena. Unlike classical digital computers in which the basic unit of computa^on, the bit, has to be in one of two discrete binary states (1 or 0), quantum computers u^lise qubits or qudits which may exist in a superposi^on of different computa^onal states. This is important for quantum chemistry applica^ons, because it enables the computer to accurately describe molecules with exponen^ally fewer qubits than bits on a classical computer. However, the current genera^on of quantum computers are vulnerable to stochas^c noise which may lead to decoherence between the different quantum states. This noise problem becomes more significant as the number of qubits and gates increase for more complex computa^onal processing. Currently available quantum computers are o1en referred to as noisy intermediate- scale quantum (NISQ) devices, in recogni^on of the significant problem of noisy qubits. The presence of noise and the current lack of complete fault-tolerance causes significant limita^ons in the types of algorithms that are executable on currently-available NISQ devices, where considera^ons such as the circuit depth or the available resources (e.g., limited number of qubits and a lack of gatefidelity) need to be taken into account. Therefore, mi^ga^on of the problems of noise is a significant technical problem associated with NISQ devices. Quantum computers have also been emulated using classical computers, and quantum compu^ng systems have been implemented using a hybrid combina^on of a classical digital computer and a quantum computer, partly because of the need to improve the reliability and scalability of quantum computers. There is interest in developing computa^onal procedures and implementa^ons of quantum computers for simula^ng and evalua^ng the physical proper^es of materials, such as energy levels and par^cle interac^ons. When using quantum compu^ng systems to simulate the behaviour of physical quantum systems, such as a molecule, there is a need for a computer- implemented method to prepare a low-energy electronic structure state and to determine the energy of the state. For the results to be useful, there is a desire for computa^onal results such as these energy determina^ons to have a level of precision referred to as ‘chemical accuracy’ (generally considered to be a level of accuracy of 10-3atomic units), but noisy quantum gates reduce the accuracy of energy measurements and discre^za^on errors arise when a computa^on 1 MJJ / 168922PCT1 is broken down into a large number of discrete steps (e.g. when simula^ng the evolu^on of a quantum system by Tro;erising ^me-dependent electronic-structure Hamiltonians). Known approaches for determining the ground state energy of a simula^on of a complex quantum system are not implementable with sufficient accuracy on currently available classical computers or NISQ computers. Summary Aspects of the present inven^on provide computer-implemented methods and quantum compu^ng systems for preparing computa^onal states represen^ng quantum states of a physical system, including performing a computa^onal evolu^on of the state and then determining physical proper^es of the system using the ^me-evolved computa^onal state. Afirst aspect of the present inven^on provides a computer-implemented method for compu^ng Hamiltonian dynamics of observables on a quantum computer to provide informa^on about the physical system represented by the Hamiltonian. A second aspect provides a method for preparing the electronic structure state with lowest energy of a physical system using adiaba^c evolu^on. A third aspect of the inven^on provides a method for determining the energy of a quantum state of a physical system using a ^me evolu^on operator that enables adiaba^c evolu^on of a prepared electronic structure state on a quantum compu^ng system. A fourth aspect of the inven^on provides a method for indirectly determining the energy of a prepared quantum state by evalua^ng the expecta^on values of a ^me-evolu^on operator. Example methods described below enable calcula^on of the energy of an evolved state with chemical accuracy, due to avoiding discre^za^on errors, with afinite circuit depth requiring fewer quantum gates than known alterna^ves. In this context, the achievement of chemical accuracy and avoidance of discre^za^on errors with ‘finite circuit depth’ refers to the fact that the number of gates in the quantum circuits does not increase unacceptably with precision improvements, as would happen in alterna^ve solu^ons relying on Tro;erisa^on. For this reason, the methods described herein are implementable on NISQ computers. According to an aspect of the inven^on, there is provided a computer-implemented method for determining the energy of a quantum state of a physical system or determining another physical property of the physical system, comprising: genera^ng a computa^onal model represen^ng the physical system, wherein the computa^onal model comprises an ini^al computa^onal state and a ^me-evolu^on operator for performing an evolu^on over ^me to prepare afinal computa^onal state; genera^ng quantum circuits for implemen^ng the computa^onal model, wherein each of the quantum circuits comprises a sequence of steps to implement the ^me-evolu^on operator, wherein each step of the sequence of steps is drawn randomly from a set of poten^al steps of the ^me-evolu^on operator; and execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system and measuring the qubits or qudits to output expecta^on values of the ^me-evolu^on operator; 2 MJJ / 168922PCT1 compu^ng an average of the expecta^on values over the generated quantum circuits; and determining a physical property of the physical system from the computed average of the expecta^on values. In an example implementa^on, each quantum circuit comprises, for each of a plurality of terms in the computa^onal model, a sequence of rota^ons of a selected gate angle applied to the respec^ve term. A quantum circuit may contain rota^ons of mul^ple different terms and different terms can be associated with different gate angles, but the sequence of rota^ons for each respec^ve term of the computa^onal model comprises a sequence of rota^ons having the same gate angle selected from a predefined set of gate angles. A1er compu^ng the average of the expecta^on values over the generated quantum circuits for each term of the computa^onal model, the computed average can be rescaled by a factor that is a func^on of the gate angle. The present inven^on is implementable by measuring the qubits or qudits of a quantum computer system and compu^ng expecta^on values as an average of measurement outcomes over several shots (i.e. realiza^ons of the same circuit) for each of a plurality of randomly- generated quantum circuits. The inven^on can be used, for example, to determine the effects on the proper^es of physical systems, including the proper^es of their electronic structure states, from chemical interac^ons that stretch the bonds within a molecule. The inven^on can be used to determine reac^on rates by compu^ng the ground state energy as a func^on of interac^ons that cause bond stretching. The inven^on can also enable a determina^on of the effect of temperature on molecules, such as determining the temperature at which molecules will dissociate, for example. As well as determining the ground state energy of a physical system or the energy of another equilibrium state, the inven^on could also be used for compu^ng quan^^es out-of-equilibrium, such as orbital occupa^on for example. The generated computa^onal model may comprise a Hamiltonian, H, which represents the states and interac^ons of a physical system. This can take account of the composi^on of molecules and approximate configura^on of atoms and the effects of orbital electrons, including building a Hamiltonian that is designed to reflect the results of experimental tes^ng of the physical system, such as experimentally tes^ng molecules for a specific objec^ve and reflec^ng that test data within an objec^ve func^on of the Hamiltonian. The evolu^on operator of the Hamiltonian allows a computa^onal determina^on of the energy of a state of the represented physical system, implemen^ng an evolu^on of an ini^al computa^onal state followed by a determina^on of proper^es of that evolved state such as the ground state energy. Computa^onal states represen^ng states of the physical system may be evolved gradually from a prepared ini^al state using quantum circuits that average to an adiaba^c evolu^on of the physical system. Expecta^on values of a ^me-evolu^on operator of the Hamiltonian may be calculated as an average of measurement outcomes over a plurality of shots of each of a plurality of quantum circuits. In an example implementa^on, the sequence of opera^ons of each quantum circuit comprises a sequence of rota^ons with a selected gate angle, for a respec^ve term of the 3 MJJ / 168922PCT1 Hamiltonian, wherein each rota^on is drawn randomly and independent of each other with a rate (corresponding to a probability) given by a func^on of the gate angle. The rota^ons for a respec^ve term of the Hamiltonian have the same gate angle, but each quantum circuit may comprise rota^ons with a different gate angle. Rota^on operators in a quantum circuit manipulate qubits to change their orienta^on, modifying the quantum state. A sequence of rota^ons can transform an ini^al state into a desired state. The gate angle is a parameter that specifies the amount of rota^on in Hilbert space that is applied by a quantum gate operator. Adjus^ng the gate angle can control the state evolu^on, and in the present inven^on a sequence of small rota^ons can be used to achieve an adiaba^c state evolu^on – evolving the state while staying in the ground state. We note here that the terms of the computa^onal model may be the terms in a Pauli string decomposi^on of a Hamiltonian. The rota^ons in this context are implemented as the exponen^al of a tensor product of Pauli matrices on different qubits ^mes a coefficient which is referred to here as the gate angle. Namely, an operator ^^^^^, with a gate angle ^, where P is a Pauli string (for example P=X_0 Y_1 Z_2). An implementa^on of the above-described aspect of the present inven^on provides a computer- implemented method for performing adiaba^c ^me evolu^on calcula^ons on quantum computers, which is free of discre^za^on errors. The method involves expressing the ^me evolu^on operator as an average over randomly generated quantum circuits which have afinite number of gates. The quantum circuits are generated by appending to the circuit, for each term in a Hamiltonian, a sequence of rota^ons of the term with afixed gate angle, drawn randomly and independently with a rate given by a known func^on of the gate angle and the coefficient of the term, during a ^me equal to the simula^on ^me. The averaged expecta^on values measured are propor^onal to the exact expecta^on value with a propor^onality factor given by a known func^on of thefixed gate angle and the Hamiltonian. A computer-implemented method for determining the energy of a quantum state of a physical system comprises: preparing an ini^al computa^onal state of the physical system (such as a representa^on of a known ground state); genera^ng a ^me-dependent Hamiltonian comprising a ^me-evolu^on operator for evolving the ini^al computa^onal state towards a target computa^onal state; genera^ng quantum circuits which each implement a sequence of gates for implemen^ng the ^me-evolu^on operator to perform an adiaba^c state evolu^on from the ini^al computa^onal state to the target computa^onal state, wherein each gate of the sequence of gates is drawn randomly from a set of poten^al gates associated with the ^me-evolu^on operator; and execu^ng the quantum circuits on the qubits of a quantum compu^ng system and measuring the qubits for a plurality of shots of each quantum circuit, to output expecta^on values of the ^me-evolu^on operator; compu^ng an average of the expecta^on values over the plurality of shots of the generated quantum circuits; and determining the energy of the target computa^onal state from the computed average of the expecta^on values. 4 MJJ / 168922PCT1 According to another aspect of the inven^on, there is provided a computer-implemented method for determining the energy of a quantum state of a physical system, comprising: genera^ng a computa^onal model represen^ng the energy of a ^me-evolved quantum state of the physical system, wherein the computa^onal model comprises a prepared computa^onal state, represen^ng a state of the physical system, and a ^me-evolu^on operator performing an evolu^on over ^me of the prepared computa^onal state, wherein the operator has a parameter that encodes the energy of the ^me-evolved computa^onal state and the operator has a parameter-dependent expecta^on value that is zero when the parameter is equal to the ground state energy of the state; genera^ng quantum circuits corresponding to the computa^onal model; execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to compute the effects of the ^me-evolu^on operator of the computa^onal model; and measuring the qubits or qudits of the quantum compu^ng system so as to determine the value of the parameter where the parameter-dependent expecta^on value is zero, and using the determined parameter value to determine the ground state energy of the ^me-evolved quantum state. A key feature of this aspect of the inven^on is an indirect method for measurement of the energy of a quantum state, especially an equilibrium state such as the ground state of the physical system or another eigenstate. This indirect measurement is much more robust to hardware noise than direct energy measurement. It involves measuring a parameter-dependent operator whose expecta^on value vanishes at a certain parameter value, to determine the parameter value corresponding to the ground state energy of the state. Instead of directly measuring the energy of the state, which would require running circuits with lower noise than the target precision, the energy of the state is encoded in the value of the parameter where this operator has a vanishing expecta^on value. Since the main effect of hardware noise is to a;enuate the signals measured, this approach of determining the point at which the operator has a zero expecta^on value has been observed by the inventors to tolerate higher levels of noise. The method of this measurement aspect can be implemented using the above-described state evolu^on followed by indirect measurement to determine the ground state energy. The computa^onal model that represents physical proper^es of the system may include experimentally-determined proper^es for an ini^al electronic structure state, followed by ^me- evolu^on of the computa^onal state and use of that ^me-evolved state in the determina^on of physical proper^es and indirect measurement. The inven^on allows a computa^onal determina^on of the ground state energy of a ^me-evolved quantum state. This can be used, for example, to determine the effects of physical interac^ons on the ground state energy of a physical system. Described below by way of example are three methods for measuring the energy of the prepared state in an efficient and noise-resilient manner, each of which has been shown to yield chemically accurate results on a 4-qubit molecule in the presence of realis^c gate noise, without the need for error mi^ga^on when performing a determina^on of the ground state energy for small molecules. 5 MJJ / 168922PCT1 Features of the above-described aspects of the inven^on may be combined. According to one aspect of the inven^on, there is provided a computer-implemented method to determine the energy of a quantum state of a physical system, by calcula^ng an expecta^on value of a parameter-dependent state evolu^on operator as an average over a plurality of quantum circuits, and indirectly measuring the energy. The method comprises: genera^ng a computa^onal model represen^ng the energy of a physical system, wherein the computa^onal model comprises an ini^al computa^onal state represen^ng a state of the physical system and an operator represen^ng an evolu^on over ^me of the physical system from the ini^al state, wherein the operator is selected to have a parameter-dependent expecta^on value that is zero when the parameter is equal to the energy of the state; ; genera^ng quantum circuits for implemen^ng the computa^onal model, wherein each of the quantum circuits comprises a sequence of opera^ons of the ^me-evolu^on operator, wherein each opera^on of the sequence of opera^ons is drawn randomly from a set of opera^ons of the ^me-evolu^on operator; and execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to compute the effects of the ^me-evolu^on operator of the computa^onal model; and measuring the qubits or qudits of the quantum compu^ng system so as to determine the value of the parameter where the parameter-dependent expecta^on value is zero, to output measurement results corresponding to the ground state energy of the ^me-evolved quantum state. This approach avoids directly measuring the energy of the state, making use of the calculated expecta^on values for a ^me-evolu^on operator applied on the ground state, by encoding the energy of the state in the value of a parameter and determining where the parameter-dependent ^me-evolu^on operator has a zero expecta^on value. The inventors have determined that this approach is less sensi^ve to system noise than known methods for measuring the energy of a state. In one implementa^on of the described method, the parameter for which the expecta^on value of the operator vanishes is directly the energy of the state. In par^cular, the inven^ve systems and methods described below have been shown to provide chemically accurate results for small molecules in the presence of realis^c gate noise, without the need for error mi^ga^on. With high-fidelity qubits and / or error mi^ga^on, the inven^on is applicable to larger molecules. In an example, the state prepara^on comprises an adiaba^c state prepara^on that generates a state corresponding to an eigenstate of the physical system, such as the ground state. The computa^onal model may comprise a Hamiltonian represen^ng the evolu^on of the physical system from a prepared state, with computa^ons involving an adiaba^c prepara^on of a state and determina^on of the ground state energy based on a measurement of observables other than a direct measurement of the energy of the state. The adiaba^c state prepara^on involves preparing a quantum system in a specific desired state, preferably star^ng with a Hamiltonian whose ground state is rela^vely easy to prepare, and then gradually transforming the ini^al Hamiltonian into a target Hamiltonian whose ground state corresponds to the desired state. The final Hamiltonian can encode the solu^on to a problem of interest. This type of adiaba^c state prepara^on has applica^ons in quantum chemistry, for tasks such as molecular simula^on and 6 MJJ / 168922PCT1 electronic structure calcula^ons, such as to determine chemical reac^on rates. An aspect of the present inven^on provides an efficient way of determining the energy of the evolved ground state of a physical system, without requiring such a high quantum circuit depth. Thus, the inven^ve approach described herein reduces the circuit depth compared to conven^onal implementa^ons of quantum phase es^ma^on (QPE). For increased efficiency on most of the currently available hardware, it is preferred to use simplified methods such as the binary search approach, arctanfit approach or Robbins-Munro approach described below. These described methods each provide a level of noise-resilience in the determina^on of the energy of an adiaba^cally prepared ground state, and demonstrate that adiaba^c approaches to state prepara^on can play a key role in quantum chemistry simula^ons, whether using noisy intermediate scale quantum (NISQ) computers or error-corrected quantum computers. The methods described herein are designed to take account of the constraints and capabili^es of exis^ng quantum compu^ng systems and include adapta^ons to exploit connec^vity between remote qubits in an ion-trapping quantum computer system. An ion-trapping quantum computer system typically has significantly lower error rate than other quantum compu^ng systems, offers flexible connec^vity between qubits (not limited to nearest neighbours connec^ons, and allowing any-to-any qubit connec^vity in some ion trapping systems), has a rela^vely lower clockspeed, displays hardware noise that is highly localized on the qubits on which gates are applied (namely, there is very low cross-talk noise), and gate-unrelated errors are mostly memory error, which acts like an approximately uniform rota^on on the qubits. These proper^es are specific to ion-trapping quantum computer systems, and do not hold for superconduc^ng quantum computer systems for example. The algorithm of this inven^on is par^cularly adapted to these ion-trapping quantum computer systems for the following reasons: (i) Chemical Hamiltonians expressed in terms of molecular orbitals and qubits involve a large number of couplings between arbitrary qubits, such that the physical system to be analyzed may be considered all-to-all coupled, and so the processing of chemical Hamiltonians is computa^onally far cheaper to implement on ion-trap devices with a built-in any-to-any connec^vity. (ii) The method described herein implements rota^ons of terms in the Hamiltonian in afixed way drawn at random for every circuit. This precludes op^mizing the gate ordering tofit the connec^vity between the qubits on other hardware systems that do not supportflexible connec^vity, because this op^miza^on would take much longer than running the circuits themselves, especially on fast quantum compu^ng systems like superconduc^ng systems. This aspect will thus be more difficult to implement on other hardware systems than ion-traps. (iii) The indirect measurement approach requires to use an ancilla qubit which is frequently acted on. On ion-trapped architectures where the ions are shu;led around different “gate zones”, this ancilla can be kept in the same gate zone of the system where entangling gates are performed, minimizing shu;ling and swapping qubits. (iv) The na^ve two-qubit entangling gate on an ion-trapping system is a ZZ- rota^on with a tuneable angle, which is valuable for the present method when implemen^ng many such small-angle rota^ons. Moreover, gatefidelity on Quan^nuum’s ion trapping devices improves for smaller angles. (v) Sources of error that are usually difficult to handle such as memory error (due to miscalibra^on of varying magne^cfield in the ion trap), modelled by random global rota^ons in the Z direc^on, have fewer effects due to par^cle number 7 MJJ / 168922PCT1 conserva^on in these chemical systems, because the sum of Z on all qubits is conserved. When implemented on a quantum computer allowing quantum gates between qubits other than their nearest neighbours (including those with all-to-all connec^vity between the qubits), the computer-implemented methods described herein can be implemented with reduced circuit depth and errors compared with quantum compu^ng systems that only support quantum gates using nearest neighbour qubits. Therefore, the described methods are op^mised for quantum compu^ng systems featuring connec^vity between non-adjacent high-fidelity qubits. An example system of this type is Quan^nuum’s H2 ion-trapping quantum compu^ng system. Quan^nuum’s ion trapping quantum computer systems enable representa^on of the evolu^on of states and the interac^ons between atomic orbitals when forming molecular orbitals in a molecule, with reduced qubit errors because of the rela^vely highfidelity of ion trap qubits. However, the inven^on is implementable on other quantum computers. An aspect of the present inven^on provides a quantum compu^ng system comprising a quantum computer having a plurality of qubits or qudits for execu^ng quantum circuits, and further comprising a controller for controlling performance of opera^ons on the quantum computer, to perform a method comprising: genera^ng a computa^onal model represen^ng the physical system, wherein the computa^onal model comprises an ini^al computa^onal state and a ^me-evolu^on operator for performing an evolu^on over ^me to prepare a ^me-evolved computa^onal state; genera^ng quantum circuits for implemen^ng the computa^onal model, which quantum circuits each comprise a sequence of opera^ons of the ^me-evolu^on operator, wherein each opera^on of the sequence of opera^ons is drawn randomly from a set of opera^ons of the ^me- evolu^on operator; execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to output expecta^on values of the ^me evolu^on operator; compu^ng an average of the expecta^on values over the generated quantum circuits; and rescaling the computed average by a factor that is a func^on of the gate angle, thereby to output a determina^on of a physical property of the physical system. Any reference to qubits herein should be interpreted as poten^ally including qubits and / or qudits, unless stated otherwise. Brief descrip^on of Figures Methods and systems implemen^ng the present inven^on are described below in detail, by way of example, with reference to the accompanying drawings in which: Figure 1A is a graphical representa^on of the precision of the determina^on of ground state energy for the current inven^on compared with a known approach to adiaba^c state prepara^on using Tro;erisa^on, showing the precision as a func^on of the dura^on of the adiaba^c state evolu^on. 8 MJJ / 168922PCT1 Figure 1B shows how the difference between a trial energy value and the ground state energy is affected by noise, but with much less influence at the point where the trial value equals the ground state energy. Figure 2A and 2B show examples of the op^mal gate angle (Fig.2A) and minimum total run^me (Fig.2B) for different values of noise. Figure 3 shows an es^mate of the total run^me to achieve chemical precision with a described binary search approach for different values of noise. Figures 4A and 4B show how the (reduced) norm of the Hamiltonian scales with the number of qubits for an algorithm implemen^ng the method of the inven^on, which is relevant for the run^me of the algorithm.. Figure 5A shows the difference between the ground state energy EGS and the determined energy E of the state adiaba^cally evolved up to a ^me T, as a func^on of T. Figure 5B shows the lowest adiaba^c evolu^on ^mes, Tmin, for which a precision of 10-3is reached (which is referred to as “chemical accuracy”), using the example of a hydrogen chain for linear and quadra^c paths. Figure 6A shows the minimum adiaba^c evolu^on ^me Tmin required to reach precision 10−3on the ground state energy, using the linear path and quadra^c path, as a func^on of the number of qubits, for a subset of molecules appearing in Figure 4. Figure 6B shows the adiaba^c evolu^on ^me as a func^on of the stretching defined as the ra^o of the bond length with the equilibrium value. Figure 7 shows the difference between the ground state energy EGSat stretching value 3 and the energy E of the state adiaba^cally prepared evolved up to ^me T , as a func^on of T for example molecules. Figure 8 is a graphical representa^on of sta^s^cal variance of state measurements as a func^on of T for different gate angles. Figure 9A shows the difference between the energy of the prepared state and the ground state energy as a func^on of adiaba^c evolu^on ^me T for different numbers of Tro;er steps, for a method relying on Tro;erisa^on. Figure 9B compares a Tro;erisa^on technique with an implementa^on of the present inven^on, by showing the number of Pauli string rota^ons required to achieve chemical precision when adiaba^cally preparing the ground state with a linear path. Figure 10 represents the amplitude of the imaginary part of expecta^on values for an operator 9 MJJ / 168922PCT1 in a state prepared using adiaba^c state prepara^on. Figure 11A shows the expecta^on value of the ^me evolu^on operator on the ground state for different trial energies, as a func^on of simula^on ^me, with posi^ve / nega^ve amplitude indica^ng over / underes^ma^on of the ground state energy. Figure 11B shows a comparison of the precision of calculated energy values using a randomised sampling technique and either an artanfit or a Robbins-Monro approach, as a func^on of the number of samples. Figure 12 shows the real part of the Loschmidt amplitude 〈0|e^itH |0〉 in the 2D Ising model at h=2 in size 3×4 with |0〉 the Z=+1 product state, comparing Tro;er, qDRIFT and the inventors’ algorithm, all with the same gate angle and Tro;er step equal to τ=0.1, and all with 105shots per point. For qDRIFT and the inventors’ algorithm, the 105shots are distributed over 103random circuits with 102shots per circuit. Figure 13A shows a noisy simula^on of a computa^onal chemistry Hamiltonian. Figure 13B shows an adiaba^c state prepara^on for noiseless simula^ons with 105circuits and τ=0.04 (circles with a dashed line , the shade indica^ng one standard devia^on), and exact value (lighter con^nuous line). Figure 14 is a schema^c representa^on of an example quantum compu^ng system comprising a hybrid combina^on of a classical digital computer and a quantum computer. Figure 15 shows a sequence of steps of a method according to an aspect of the inven^on. Figure 16 shows a sequence of steps of a method according to an aspect of the inven^on. Detailed Descrip^on 1. Introduc^on For chemists, there is interest in determining what reac^ons can occur in a physical system, and at what temperature and rate. A simple example is to determine the point at which water molecules will dissociate into hydrogen and oxygen. A determina^on of the energy of the ground state of a molecule or another quantum system is an important part of this determina^on. Quantum computers can be used for computa^onal chemistry, to determine the energy or other proper^es of a quantum system from a mathema^cal representa^on of the states of the quantum system. Let us assume that we have prepared a computa^onal state in a quantum compu^ng system, with sufficient accuracy that it corresponds to an eigenstate of the physical system. We must now evaluate the energy of the state – for example determining the ground-state energy of a molecule 10 MJJ / 168922PCT1 with high accuracy. For example, we may wish to determine a ground state energy with an accuracy within 1.6 x 10-3atomic units (1.6 mH), which is some^mes referred to as a threshold for “chemical accuracy”. The minimum energy of a quantum system may be influenced by bond lengths, and it is possible with advanced quantum computers such as Quan^nuum’s H2 trapped-ion quantum computer system to take account of varia^ons such as a stretched molecular geometry when compu^ng the energy of a ground state of a quantum system. The ability of the present inven^on to perform determina^ons of the energy of quantum states of a physical system, such as a ^me-evolved ground state that accounts for stretching of bond lengths, is facilitated by the all-to-all connec^vity of high-fidelity qubits in Quan^nuum’s trapped ion quantum computers. Chemical systems typically require highly correlated long-range Hamiltonians when performing computa^onal chemistry on the qubits of a quantum compu^ng system, and discre^sa^on to break the complex computa^ons down into smaller computa^ons (Tro;erisa^on) has resulted in too many errors when performed on systems that require all interac^ons to be exclusively between nearest neighbour qubits. An ion trapping quantum computer system is well-adapted to determina^ons of the proper^es of complex chemical systems, and the computer- implemented methods described herein are well-adapted to an ion-trapping system featuring all-to-all connected qubits. Described below are computer-implemented methods and quantum compu^ng systems for determining proper^es of a physical system, in par^cular for determining the energy of an equilibrium quantum state such as the ground state, using adiaba^c evolu^on of an ini^al computa^on state towards a target state. The described methods and systems can provide results within the currently-accepted threshold for chemical accuracy, at least for some small molecules, in a noisy quantum computer system without error correc^on. The methods have been shown to yield chemically accurate results on a 4-qubit molecule in the presence of realis^c gate noise, without the need for error mi^ga^on other than simple symmetryfiltering of the shot outcomes. The described methods and algorithms involve noise-resilient approaches for determining the energy of a prepared state, which is par^cularly advantageous when combined with adiaba^c state prepara^on and evolu^on. This has been achieved without discre^sa^on errors hea^ng the state, and with far fewer gates than Tro;erisa^on. Represen^ng low-energy states of the electronic structure Hamiltonian is an important step in quantum computa^onal chemistry. Ini^al proposals for the use of quantum computers for this task have used Quantum Phase Es^ma^on (QPE) to project high-overlap Hartree-Fock wavefunc^ons into the ground state with high probability [2]. The high cost of the coherent ^me- evolu^on required, as well as the existence of strongly mul^reference ground states (i.e. molecules where the Hartree-Fock state has low overlap with the ground state) has triggered interest in how to efficiently prepare a low-energy electronic structure state on a digital quantum computer. 11 MJJ / 168922PCT1 Heuris^c approaches, most notably the varia^onal quantum eigensolver (VQE), have been devised and carried out on noisy devices [3-5]. While inspired by adiaba^c evolu^on [6], these methods abandon the theore^cal guarantees of the adiaba^c approach in favor of parameterized circuits which need to be op^mized. For the classical computer, this replaces the quantum simula^on problem with the usually hard problem of non-linear op^miza^on in a generically hos^le cost landscape [7-11]. In contrast, adiaba^c approaches to the state prepara^on problem have received considerably less a;en^on, despite more theore^cal guarantees and fewer hidden costs than VQE [2, 12-18]. A major reason for this is that the adiaba^c evolu^on is usually assumed to be implemented via Tro;erisa^on, in part because more sophis^cated methods suffer from extra overheads [19-26], especially when simula^ng ^me-dependent Hamiltonians [20, 21, 24, 27-29]. Tro;erising the electronic structure Hamiltonian, which contains ^(^^) terms where ^ is the number of orbitals, then requires extremely large numbers of gates. What’s more, discre^sing a con^nuous adiaba^c path into piecewise constant Tro;er steps necessarily leads to discre^sa^on errors (Tro;er errors), which effec^vely heat the quantum state. This is par^cularly problema^c in quantum chemistry applica^ons, where extreme precision is required to have predic^ve power on chemical reac^on rates. An example computer-implemented method for implemen^ng the present inven^on employs a randomized ‘TETRIS’ algorithm (Time Evolu^on Through Random Independent Sampling) for Hamiltonian evolu^on, which combines adiaba^c state prepara^on with random sampling. TheTETRIS algorithm is an algorithm that computes Hamiltonian dynamics of observables 〈ℳ(^)〉 = on digital quantum computers without discre^za^on error even with afinite number of gates, for any Hamiltonian ^ and unitary ℳ . The expecta^on value 〈ℳ(^)〉 is obtained as the average over random circuits drawn from a known distribu^on, mul^plied by a known factor. One can choose freely the angle ^ of the rota^ons entering the circuit, only modifying the mul^plica^ve factor and not the precision of the result. The inventors have determined that this allows a decrease of the gate count, at the cost of an increased number of shots on a quantum computer, yielding a natural noise-mi^ga^on technique. Op^misa^on of the gate angle is described in sec^on 3.4 and Appendix 1. The op^mal choice of gate angle yields a shot overhead of order ^(1) with a gate count per circuit ^ where ^ is the 1-norm of the Hamiltonian, without any dependence on the precision desired, contrary to all previous algorithms. Since the gate count depends only on ^, the algorithm is par^cularly adapted to non-sparse Hamiltonians. It also straigh^orwardly generalizes to ^me-dependent Hamiltonians without any overhead. Thus, the method implements adiaba^c evolu^on without hea^ng and with far fewer gates than Tro;erisa^on. This avoids the discre^za^on errors that typically arise when Tro;erising ^me-dependent electronic structure Hamiltonians, and therefore removes one of the main obstacles to the use of adiaba^c state prepara^on for computa^onal chemistry applica^ons. Addi^onally, the method requires a circuit depth that is independent of the desired precision ^, contrary to many algorithms thatrequire an infinite depth when ^ → 0. Hence, by applying randomly gates on the ini^al stateaccording to a gate crea^on process (as described below in sec^on 3.2) , and mul^plying the 12 MJJ / 168922PCT1 result by a known amplifica^on factor, one achieves an energy determina^on with zero discre^za^on error while havingfinite-depth circuits. Figure 1A provides a graphical representa^on of the performance of the inven^on (including achievement of chemical accuracy) compared with a known technique based on Tro;erisa^on. Figure 1B is a conceptualfigure summarizing one aspect of the inven^on. Figure 1A represents the accuracy of the determined energy of the ground state prepared adiaba^cally, as a func^on of the total adiaba^c ^me evolu^on. A Tro;er decomposi^on leads to hea^ng, precluding reaching chemical accuracy. In contrast, the solu^on proposed herein implements the exact adiaba^c evolu^on and is free of any discre^za^on error hea^ng. Figure 1B shows theimaginary part of a Hamiltonian operator on a prepared state, as a func^on of ^ −^!", atfixed arbitrary #. The main effect of hardware noise is tofla;en the curve of Figure 1B, which does not affect the point where the curve vanishes. This zero point is where E matches the ground state energy ^!". The data for Figure 1A was obtained from an H%hydrogen chain and the data for Figure 1B was obtained from an H^hydrogen molecule, both in a STO-3G basis, see Sec^on 5 and below. A new computer-implemented method is described herein using an implementa^on of the randomised “TETRIS" algorithm (Time Evolu^on Through Random Independent Sampling) for Hamiltonian simula^on that was described in E. Granet and H. Dreyer, arXiv:2308.03694 (2023), 10.48550 / arXiv.2308.03694 [1]. That paper is incorporated herein by reference and included as Appendix 1. The TETRIS algorithm takes as input a Hamiltonian ^ decomposed, a unitary observable ℳ, a^me ^ and an ini^al state |&(0)〉, and outputs 〈&(^)|ℳ|&(^)〉. It takes 0 < ^( ≤ * / 2 as -parameters. The algorithm consists of the following steps:firstly, prepare the system in the ini^alstate |&(0)〉 . Secondly, for each ... , -} , draw an integer 4( from a Poisson|5 distribu^on with parameter^|^ 678|^^|. Then draw 4( real numbers ^(^)( , where 9 ∈ {1, ... , uniformly at random between 0 and ^. These : ≡ ∑=(>? 4( numbers are collected into a set@ . Thirdly, deduce then the sequence A?, ... , AB ∈ {1, ... , -} such that the index .corresponding to the C-th smallest element of @ is AD. For 4 = in this order, applythe rota^on ^^^EF6G( (5EF)HEFon the system. Fourthly, apply the observable ℳ on the system. Fi1hly, repeat the second and third steps with ^(replaced by −^(. Sixthly, measure the overlap with the ini^al state and divide the result by an a;enua^on factor. On a quantum computer, this step requires to perform a Hadamard test. Then repeat thefirst to sixth steps a number of ^mes and average the results. Sec^on 3B of this patent specifica^on also discloses these steps of the TETRIS algorithm. When measuring the overlap with the ini^al state, one has to choose a number of shots I to do per configura^on, i.e. how to distribute the resources between number of configura^ons and accuracy of each configura^on. If we neglect compila^on^me, the op^mal number of shots can be seen to be I = 1

[0117] .This TETRIS algorithm has the following features: (i) the cost depends not on the number of terms in the Hamiltonian but on the interac^on norm ^ (which we will introduce shortly) which is 13 MJJ / 168922PCT1 bounded by the 1-norm and generically has a scaling between ^(^) and ^(^K) depending on the system and the decomposi^on considered [30-32], (ii) the number of gates can be chosen at will at the cost of introducing sampling overhead, (iii) ^me-dependent Hamiltonians can be simulated at no extra cost, (iv) there are no parameters to be op^mised, (v) the output is free from discre^za^on (Tro;er) error and (vi) in prac^ce, the required number of gates not onlyscales advantageously but also comes with a prefactor known to be < 1.This algorithm removes thus one of the main obstacles to adiaba^c state prepara^on (ASP) for chemistry applica^ons. The inventors have shown that states that hold informa^on about the ground state energy of small molecules within chemical accuracy can be prepared adiaba^cally in this way with circuits containing only tens to hundreds of two-qubit gates. For example, witharound ∼ 10K two-qubit gates, one can prepare the ground state of molecules like LiH or ahydrogen chain H^at equilibrium within chemical precision. Thisfinding, however, introduces a new challenge: State prepara^on is only useful to the extent that observables (most notably the energy) can be extracted from the state. The distribu^on of states created by the TETRIS algorithm agree with the exact ^me-evolved state only up to the first moment. Namely, it produces unitary operators M that sta^s^cally average to the ^me evolu^on operator ^^^^, but quan^^es quadra^c in M like MN^M with some operator ^ do not average to ^^^^^^^^^^. This renders imprac^cal many techniques developed for measuring the energy when the state is prepared using e.g., VQE. Our second main contribu^on is to overcome this challenge by developing three approaches to efficiently measure arbitrary observables in a state prepared by the TETRIS method. Remarkably, the measurement techniques are resilient to noise, and we show that the combined programme of adiaba^c state prepara^on plus measurement yields chemically accurate results on a 4-qubit problem using100’000 < (1e − 3)^^ shots on a noisy quantum computer with errors modeled by depolarisingnoise on two-qubit gates withfidelity 99.82%

[0033] and without error mi^ga^on apart from trivial symmetryfiltering. To summarise, the two-step method we propose is depicted in Figures 1A and 1B and consists of: • an adiaba^c prepara^on of the ground state Q|i.9〉 where |i.9〉 is some ini^al state and Q the operator implemen^ng the adiaba^c path. Using the algorithm of [1], this step can be implemented without any discre^za^on error hea^ng. • an energy measurement to determine the energy of the state Q|i.9〉. We propose three different related methods: a binary search approach, an arctanfit approach, and a Robbins- Monro approach. These three methods possess some built-in noise resilience. This patent specifica^on is organized as follows. We start in Sec^on 2 with a quick descrip^on of the setup encountered in quantum chemistry applica^ons of quantum compu^ng. In Sec^on 3, we present the algorithm of [1] to implement ^me evolu^on under a ^me-dependent Hamiltonian. We discuss how to best choose the parameters of the algorithm and give some run^me es^mate for ASP. In Sec^on 4 we then discuss how to best measure the energy of the state that we prepared adiaba^cally. We introduce three different approaches for the 14 MJJ / 168922PCT1 computa^on of the energy. Then in Sec^on 5 we do a resource es^mate of the algorithm to reach chemical precision in mul^ple chemical systems. In par^cular we inves^gate the scaling with system size of the ^me that is required to perform ASP. Then in Sec^on 6 we present some noisy simula^ons of the algorithm. Finally, Sec^on 7 provides a summary of thefindings and conclusions. 2. Setup and nota^ons Quantum chemistry aims at compu^ng the ground state energy of molecules

[0034] . These are described by the following Hamiltonian where _^ , D̀ , :^ , a^ are respec^vely the posi^ons of the nuclei, the posi^ons of the electrons,the masses of the nuclei and the charges of the nuclei, wri;en here in atomic units. The movement of the nuclei can usually be neglected with the Born-Oppenheimer approxima^on

[0034] , leaving only electronic degrees of freedomD̀and a parametric dependence of the ground state energy in the nuclei posi^ons _^. The Hilbert space associated to this Hamiltonian is infinite-dimensional, and has to be discre^zed in some way to be implemented on a computer (whether classical or quantum). Approaches referred to as “first quan^za^on" discre^ze space [35-37], while “second quan^za^on" methods project the Hilbert space onto linear combina^ons of afinite number of well-chosen states that approximate equilibrium configura^ons, called basis set. In this patent specifica^on we will consider the second quan^za^on approach, which is the most studied. Here, the Hamiltonian is (approximately) wri;en as where ^ is the number of orbitals (which depends on the basis set), ℎcd , ℎcd[^ are realcoefficients, and fc, fNc are fermionic operators sa^sfying canonical an^commuta^on rela^ons{fc, fNd} = gc,d and {fc, fd} = 0. To be implemented on a quantum computer, this second-quan^zed Hamiltonian must be wri;en in terms of Pauli matrices h, i, a . This can be donethrough different mappings [38-40]. We will consider the widely used Jordan-Wigner transforma^on Once the Hamiltonian ^ is wri;en in terms of Pauli matrices, our objec^ve is to determine its ground state with some precision ^. In chemistry applica^ons, this precision is very o1en set to1.6 ⋅ 10^K = 1.6mH, called chemical accuracy, which corresponds approximately to the accuracyachieved in experiments

[0041] . We will take the precision to be 10^Kfor simplicity. We note that in this patent specifica^on we will not be concerned with the accuracy of the basis set: the precision we wish to achieve is with respect to the ground state of the (approximated, truncated) 15 MJJ / 168922PCT1 Hamiltonian ^ expressed in afinite basis set, not with respect to original infinite-dimensional Hamiltonian. This is referred to as chemical precision

[0042] . 3. Adiaba^c prepara^on 3.1. Adiaba^c path In this Sec^on, we present thefirst step of our proposal, which is the adiaba^c prepara^on of the ground state of ^. Adiaba^c State Prepara^on (ASP) relies on the adiaba^c theorem of quantum mechanics, which roughly states that if parameters of the Hamiltonian are varied slowly enough, a system ini^ally in the ground state will remain in the ground state of the ^me-dependent Hamiltonian at all ^mes [43-45]. Specifically, one considers a ^me-dependent Hamiltonian ^(l)for 0 ≤ l ≤ 1, and an ini^al state |i.9〉, such that:(i) |i.9〉 is the ground state of ^(0), (ii) thefinal ^me Hamiltonian ^(1) is the target Hamiltonian ^, and (iii) for all l , ^(l) is gapped (i.e. there is a clear separa^on or gap between the ground state energy and thefirst excited state energy).Then, evolving |i.9〉 with the ^me-dependent Hamiltonian ^(^ / @) from ^ = 0 to ^ = @(where t is the actual physical ^me, and T the total simula^on ^me, and u is equal to t / T), one prepares a state Q(@)|i.9〉 that becomes closer to the ground state of ^ as @ grows larger. T is a parameter that controls the speed of the adiaba^c evolu^on, but which isfixed for every simula^on run. To implement an exact adiaba^c evolu^on, a large T should be chosen. The convergence of the energy of Q(@)|i.9〉 is typically polynomial in 1 / @, but can be exponen^al in @ if the path is smooth enough [46-48]. The ^me needed then scales as the inverse square of the smallest gap of ^(l) along the path

[0049] . The operator Q(@) that implements this ^me evolu^on under a ^me-dependent Hamiltonian can be wri;en with the following ^me-ordered exponen^al d^u. (4) There are many valid paths ^(l) that can be considered for ASP. In our case, we will assume the simple following sebng that is well adapted to quantum chemistry Hamiltonians. We split ^ into two terms ^= ^v + ^J , (5)where the ground state of the background Hamiltonian ^vcontains only a Pauli matrix terms. More generally, one could consider any Hamiltonian ^vbut the adiaba^c state prepara^on method is most helpful if the ground state of this Hamiltonian is rela^vely easy to prepare on a quantum computer (i.e. the prepara^on of the ground state of ^vshould require far fewer gates than the adiaba^c evolu^on itself). We decompose then the interac^on Hamiltonian ^Jthat contains the remaining terms into Pauli strings 16 MJJ / 168922PCT1with x( ∈ ±{z, h, i, a}⊗b and f( real coefficients. Up toflipping x( into −x( , we canassume f( ≥ 0. For nota^onal convenience, we will denote as well where here x(are strings of a Pauli matrices at some sites ^mes a ± sign, and with - the total number of terms in the Hamiltonian ^. Then we take the ini^al state |i.9〉 as the ground state of ^v, which can always be prepared efficiently on a quantum computer as it is a product state (i.e. a tensor product of states on every qubit; it can be prepared efficiently as one can just apply one-qubit gates on every qubit, star^ng from |000>). For a given weight func^on (or path)~(l) ≥ 0 such that ~(0) = 0 and ~(1) = 1, we then set This adiaba^c path could be refined by e.g. sebng a different weight ~(l) for every Pauli string in ^J. However, the adiaba^c path can be implemented with a single weight func^on for all terms and this simpler implementa^on is the example described in this patent specifica^on. 3.2. Algorithm for implemen^ng ^(^)Lg The operator Q(@) can be implemented exactly with afinite circuit depth on a quantum computer, using the randomiza^on algorithm introduced in Appendix 1 of this patent specifica^on and in [1]. To present this algorithm ^, we define as well as ^^?(l) the reciprocal func^on of the increasing func^on ^(l), namely such that^^?(^(l)) = l for all l. We will denote which is of order ^(1) . The algorithm takes as a parameter a gate angle 0 < ^ < * / 2. Itconsists of the following steps. 1. Ini^alize the state of the system |&〉 to |i.9〉. 2. For . = 1, ... , -J, draw an integer 4( ≥ 0 from a Poisson distribu^on with5^^r parameter678^. 3. For . = 1, ... , -J, draw 4( real numbers ^̃^,( (with 9 = 1, ... , 4(), independentlyand uniformly at random between 0 and ^. For every 9, ., set ^^,( = @ ^^?(^̃^,().4. For . = -J + 1, ... , -, draw an integer 4( ≥ 0 from a Poisson distribu^on with5^r parameter678^. 5. For . = -J + 1, ... , -, draw 4( real numbers ^^,( (with 9 = independently and uniformly at random between 0 and @. 6. Find the sequence such that17 MJJ / 168922PCT1 7. For 4 = 1, ... , :, apply the operatorexp^9^x(F^ (12) on the state |&〉. Let us denote M the random unitary operator that is generated with this process. The result of [1] is that where ^ denotes the sta^s^cal average with respect to M, and with ^J, ^v the 1-norm ofthe interac^on / background Hamiltonian We will call a^enua^on factor the term In each random realiza^on M , there are in average5^^r^^^^678^gates ^in the circuit for . =1, ... , -J , and 5^r678^ gates ^^^^^ for . = -J + 1, ... , - . Let us denote ^( the number of two-qubit gates that is required to implement ^^^^^. Then the average number of two-qubit gates in the circuit is which isfinite even though the exact adiaba^c prepara^on is implemented on average. 3.3. Improvement of the algorithm for implemen^ng ^(^)Lg The algorithm of Sec^on 3.2 possesses the following improvement in the case where the ^me evolu^on with respect to the background Hamiltonian ^vcan be implemented with just afinite number of gates, namely when ^^^^^can be decomposed exactly as afinite product of gates on the quantum computer. This improvement is called the background Hamiltonian technique [1]. It consists in applying con^nuously the ^me evolu^on with respect to ^v, and randomizing only the ^me evolu^on with respect to the interac^on Hamiltonian . The upside is that the a;enua^on factor ^ involves only ^Jthe 1-norm of the interac^on Hamiltonian. This improved algorithm takes as a parameter a gate angle 0 < ^ < * / 2, and consists ofthe following steps: 1. Ini^alize the state of the system |&〉 to |i.9〉. 2. For . = 1, ... , -J, draw an integer 4( ≥ 0 from a Poisson distribu^on with5^^r parameter678^. 3. For . = 1, ... , -J, draw 4( real numbers ^̃^,( (with 9 = 1, ... , 4(), independentlyand uniformly at random between 0 and ^. For every 9, ., set .18 MJJ / 168922PCT1 4. Find the sequence such that 5. For 4 = 1, ... , :, apply the operator on the state |&〉, with ^^^,(^ ≡ 0.6. Apply the state Let us denote M the random unitary operator that is generated with this process. The result of [1] is that where ^ denotes the sta^s^cal average with respect to M . In that case, we will call a;enua^on factor the term ^= ^^^^8(^ / ^)^r^w . (20)In each random realiza^on M , there are in average5^^r ^^^678^gates ^^in the circuit. Let us denote ^(the number of two-qubit gates that is required to implement ^^^^^. Then the average number of two-qubit gates in the circuit is which is againfinite. 3.4. Choice of gate angle 3.4.1. Op^mal gate angle on a noiseless quantum computer Figures 2A and 2B show the rela^onship between op^mal gate angle (Fig.2A) and minimum total run^me (Fig.2B) for different values of noise. Figure 2A shows the op^mal gate angle ^∗andFigure 2B shows the minimal total run^me _(^∗) as a func^on of @^J , with ^ = 1, for differentvalues of noise `^. Total run^me corresponds to the number of shots ^mes the number of two- qubit gates per shot, to get a precision ^(1) on the result. The total run^me to get a precision ^(^) on the result i _ The gate angle ^ is here a free parameter of the algorithm, which does not change the precision on the result. Increasing ^ reduces the number of two-qubit gates -r^^, so reduces the run^me of every random circuit. However, it also makes the a;enua^on factor ^ smaller. This a;enua^on factor increases the number of shots required to reach a certain precision. Indeed, because we have to divide the amplitude measured on the quantum computer by ^, the number of shots required to have a precision ^ on the result is (^^)^^. We elaborate on this fact in Sec^on 5.3 below. This means that for the same precision we have to run more shots of the same circuit as ^ increases. Hence, the total run^me atfixed precision, i.e. the product of run^me of 19 MJJ / 168922PCT1 each circuit and number of shots per circuit, has a non-trivial behaviour with the gate angle ^ and reaches a minimum at some op^mal ^∗. This op^mal gate angle ^∗moreover depends on the task that we wish to fulfil. Let us consider for example the problem of measuring an observable ^ within the adiaba^cally evolved state Q(@)|i.9〉. Using the above protocol, we can write where the circuits M?, M^ are drawn independently from the previous protocol. Importantly,the backward and forward propaga^ons that are randomly drawn do not (at least not necessarily)coincide M ≠ M , so the amplit N? ^ ude 〈i.9|M^^M?|i.9〉 has to be measured explicitly on thequantum computer with for example a Hadamard test. This means that both M?and M^have to be implemented explicitly, so that the total number of two-qubit gates (not taking into accountthose possibly entering ^) is 2-r^^. Each of the two averages over M?, M^ contributes with ana;enua^on factor ^, so that the total a;enua^on is ^^. On a perfect quantum computer, one thus has to run ^^^many circuits to obtain a precision ^(1) on the result, and ^^^^^^to obtain a precision ^. Hence the total number of two-qubit gates to run (across different circuits)to get precision ^ is _(^) / ^^, with _(^) = 2-r^^^^^. Explicitly, this factor _(^) is We recall that _(^) / ^^is the total number of two-qubit gates to run to reach precision ^, but each circuit has only 2-r^^two-qubit gates, independently of ^. This total run^me reaches a minimum at some ^∗ . Imposing  ^_(^∗) = 0 , wefind atlarge @^J[1] ^∗ = ?^^r^w , _ Intui^vely, in absence of noise, one should choose the largest possible ^ to reduce the number of two-qubit gates in each circuit, while s^ll keeping an a;enua^on factor ^ of order ^(1). Thisgives ^ = ^(1 / (@^J)).3.4.2. Op^mal gate angle on a noisy quantum computer This “total run^me" count holds for a perfect, noiseless quantum computer where every gate can be executed perfectly. However, in any hardware in the near future, noise cannot be neglected and is a crucial aspect to take into account in quantum compu^ng algorithms. We are going to make the assump^on that because of the imperfect hardware, the measured expecta^on valueis mul^plied by ^^[ with ` > 0 for every two-qubit gate present in the circuit. This ^^[ iscalled the effec^ve two-qubit gatefidelity. This noise model is of course a crude approxima^on of actual hardware noise, which is much more complex and pla^orm-dependent. However, it is known to give a roughfirst es^mate

[0033] and there are several noise mi^ga^on techniques such as Probabilis^c Error Cancella^on that can convert any noise occurring in the hardware into this kind of global depolarizing noise [50,51]. We note that even with quantum error correc^on, the 20 MJJ / 168922PCT1 noise level ` would not be exactly equal to 0, so this discussion is also relevant beyond the NISQ era. As long as `^(is small (if `^(is not small a single gate ^^^^^would be too noisy to implement), the noise can then be assumed to mul^ply the measured amplitude by a factor ∼¢ = ^^[=£¤¥. This has the same effect as the a;enua^on factor ^ and increases the number ofshots required to reach a given precision. Hence, the total run^me with a noise parameter ` is Let us write an equa^on for the gate angle ^∗ that minimizes _(^) . The quan^ty ¦ =tan(^∗ / 2) sa^sfies the quar^c equa^on If we set ` = 0, at large @^J we recover the previous noiseless result. In the presence of noise` > 0, the behaviour of ^∗ at large @^J is completely different. Wefind at large @^J In the presence of noise, the argument of the exponen^al appearing in _(^) cannot be keptconstant when @^J → ∞. The op^mal gate angle depends only on the noise factor `, and therun^me is exponen^al in @^J, with an exponent propor^onal to√`. In Figure 2, we show the behaviour of the op^mal gate angle ^∗and the corresponding minimal run^me _(^∗) as a func^on of @^J, for different amounts of noise ` . To give concretenumbers, ` ≈ 2 ⋅ 10^K for current state-of-the-art two-qubit gates

[0033] , while the averagenumber of two-qubit gates per rota^on ^ is roughly ^ ≈ ^ / 2 where ^ is the number ofqubits, see Figure 4 below. 3.5. Techniques for reducing the norm «¬Lg In terms of system size, the cost of the algorithm depends only on the 1 -norm ^Jof the interac^on Hamiltonian. It differs from a Tro;er decomposi^on that depends on the number of terms in the Hamiltonian, and also from Linear Combina^on of Unitaries techniques, as well as other randomiza^on techniques such as qDRIFT, which depend on the 1 -norm of the en^re Hamiltonian. Contrary to the 2 -norm, the 1 -norm is basis-dependent. There can thus be decomposi^ons of the same Hamiltonian that have different interac^on norms ^J. Since the cost of implemen^ng the ^me evolu^on for ^me ^ is a func^on of ^^J, it is valuable to use a decomposi^on that reduces this norm ^J. There exist different techniques to decrease the 1 -norm. In par^cular, in the context of chemistry applica^ons, one can perform a unitary transform on the orbital basis that minimizes the 1-norm, instead of minimizing the energy of the Hartree-Fock state

[0032] . This classical pre- processing of the Hamiltonian scales polynomially with the number of orbitals. According to

[0032] , 21 MJJ / 168922PCT1 it can for example modify the scaling with system size of the 1 -norm of the Hamiltoniancorresponding to a hydrogen chain, which then scales as ∼ ^?.^ with ^ the number ofhydrogen atoms, instead of ∼ ^^.? in the Hartree-Fock op^mized basis.Another cheap way of reducing the 1 -norm is to make use of exact symmetries in the Hamiltonian. It is well-known that symmetries can be used to reduce the number of qubits, with a process called tapering

[0052] . However, it does not necessarily reduce the 1 -norm of the Hamiltonian, which is the only relevant system-size-related parameter in the TETRIS algorithm. Instead, given a conserved quan^ty ­ , i.e. an operator that commutes with the Hamiltonian[^, ­] = 0, one can always add a mul^ple of ­ to form a new Hamiltonian ^′^′ = ^ − ®­ , (28)that has the same eigenstates as ^ with eigenvalues trivially shi1ed by their eigenvalue on −®­ . In the quantum chemistry applica^on case, par^cle number -¯°[^is a conservedquan^ty. Wefind that sebng ­ = -^¯°[^ the square of the number of par^cles, and taking ®so as to minimize ^J, can significantly reduce the norm ^J. When using the Jordan-Wigner transforma^on for the encoding, onefinds in that case that the op^mal choice of ® is given by the median value of the coefficients f(in front of Pauli strings with exactly two a ’s. In the molecules that we consider below in this patent specifica^on, we systema^cally use this trick to reduce the norm of the Hamiltonian. 4. Measuring the energy 4.1. Generali^es The algorithm that we presented in Sec^on 3 enables us to prepare (an approxima^on of) the ground state of the Hamiltonian |&(@)〉 = Q(@)|i.9〉 . (29)We now need to decide how to best measure this state and retrieve informa^on in order to fulfill a certain task. In our case, we are interested in evalua^ng the energy of the state that we prepared ^(@) = 〈&(@)|^|&(@)〉 . (30)The simplest approach to compute ^(@) is to measure individually every Pauli string in the Hamiltonian ^. A Pauli string with a small coefficient f(requires fewer shots to obtain a certain fixed precision on the total energy. At afixed total number of shots, the op^mal strategy is to spend a number of shots to measure x(that is propor^onal to f([32, 53]. This way, in order to have a precision ^ on the result, the number of shots :"that is required is where ^ is the 1-norm of the total Hamiltonian 22 MJJ / 168922PCT1 This approach has two shortcomings. Firstly, it is very sensi^ve to noise on an imperfect hardware. With this approach, obtaining a precision of order 10^Kwould inevitably require to bring the noise in the signal below 10^K. Secondly, with this approach, one necessarily computes all the bits in the binary decomposi^on of the energy ^(@), including those that can be obtained classically by exact or approximate methods. To make a be;er use of scarce quantum compu^ng resources, one would like to be able to compute only the most precise bits that are beyond reach of classical methods. Moreover, the noise will not affect the bits that are easy to compute classically, and will completely blur the bits that require quantum computa^on. 4.2. Measuring the energy through ^me evolu^on 4.2.1. QPE techniques In order to explain the present improved method to compute the energy of the state we briefly review Quantum Phase Es^ma^on (QPE) [54, 55] below, and then describe an op^misa^on. QPE is an algorithm to measure the eigenvalue of an eigenvector |&〉 of a unitary operator M , that requires only ^(log1 / ^) shots and a circuit depth ^(1 / ^) to reach a precision ^. In our case we could apply QPE to measure the energy of the state |&〉 assumed tobe the ground state, by taking M = ^^^ . The algorithm works as follows. It takes as input aunitary operator M, an eigenvector |&〉 and a target precision ^ = 2^( for some integer .,and returns . bits of the binary decomposi^on of ¦ where ^^^µ¶is the eigenvalue of M . Firs Fty, it prepares . ancilla qubits in the state |+〉. Then it performs sequen^ally . controlled-M^ opera^ons between the system and the ancillas for 4 = 1, ... , ., followed by an inverseQuantum Fourier Transform (QFT) on the ancillas. Finally, measuring the ancillas exactly provides the bits of the binary decomposi^on of ¦. What makes QPE able to yield a dras^c reduc^on of the number of shots compared to measuring all the terms in ^ is the ^me evolu^on. Measuring ^^^does not have a notable precision change compared to measuring ^. However, if we measure the eigenvalue of ^^^^instead of^^^ for some # > 0, we only need a precision #^ on the eigenvalue of ^^^^ in order to havea precision ^ on the eigenvalue of ^^^(or more precisely on its value modulo 2* / #). If onetakes # = 1 / ^, one only requires ^(1) shots to reach a precision ^. To be more concrete, wecan describe the following “itera^ve" version of QPE [56, 57]. Let us assume that 0 ≤ ¦ < 1 theeigenvalue of ^ on |&〉 has an exact binary decomposi^on ¦ = 0. · .. ¹? . ·( =bits ·^ ∈ {0,1}. If we measure we obtain a measurement outcome +1 with probability 1 if ·( = 1 , and −1 withprobability 1 if · ¹^( = 0. Then, we shi1 ^′ = ^ − If we measure23 MJJ / 168922PCT1 we obtain +1 with probability 1 if ·(^? = 1, and −1 if ·(^? = 0. Proceeding repeatedly,we can measure all the bits ·?, ... , ·(. The downside of this “itera^ve" version of QPE is that oneneeds to have an exact eigenstate of ^, and not an approxima^on, whereas QPE would succeed for general states with a probability propor^onal to the overlap of the state that we have with the target eigenstate. However, the upside of this approach is that one only requires to run . separate circuits each containing only one controlled ^me evolu^on, instead of one circuit containing . sequen^al controlled ^me evolu^ons

[0056] . It reduces thus the circuit depth, which is valuable for current and near-term devices where gates have a moderate level of noise. 4.2.2. Binary search approach The previous “itera^ve QPE" approach reduces the circuit depth compared to the original QPE. However, the ^mes during which one needs to evolve the system are s^ll large, of order 10Kto get chemical precision. On current-day devices, these very deep circuits are not feasible. In fact, the approach can be further simplified. When measuring (33) to determine the .-th bit of the energy, one does not necessarily need to do a ^me evolu^on up to ^me 2(^?. Assuming knowingthe bits ·?, ... , ·(^?, in order to determine the bit ·(, one only needs to ^me evolve sufficientlylong and take enough shots to be able to dis^nguish whether the eigenvalue a1er removing thebits ·?, ... , ·(^? is below or above 2^(^?. This observa^on suggests the following approach.Instead of seeking to measure ^, we wish to answer the following ques^on: Given an energy ^ and an eigenstate|&〉, is its eigenvalue below or above ^? ( By performing a binary search, one only needs to answer ^(log1 / ^) ques^ons (35) to locate with precision ^ the energy of the state |&〉. Moreover, each answer to (35) provides one bit of informa^on about the ground state. Contrary to measuring ^ directly, here we can choose which bit of informa^on we want to measure. Each of these yes / no ques^ons (35) can be answered by measuring the quan^ty We will call the ^me # “central ^me", to dis^nguish it from the adiaba^c prepara^on ^me @.Let us denote g = ^ − ^!" where ^!" is the energy of the state |&〉 that we are looking for.To answer ques^on (35), one only needs to decide whether Å(#) is posi^ve or nega^ve when0 < # < µThe actual value of Å(#) beyond its sign is irrelevant.To that end, we wish to choose # so as to maximize the measured value of Å(#). This would both (i) minimize the number of shots required to claim that Å(#) is either posi^ve or nega^ve, and (ii) maximize the resilience to noise, as a larger Å(#) would make it less likely to have its signflipped because of noise. We will take into account the effect of noise in determining the op^mal choice of the central ^me # . Under the same assump^ons as in Sec^on 3.4.2, the measured ÅÇÈ^(#) can be wri;en as 24 MJJ / 168922PCT1 Å(#) = Ésin ^^^ÇÈ^ (#g)^ , (37)where is a ^-dependent factor that takes into account the adiaba^c prepara^on of |&〉, and where l depends on the gate angle ^ as ^ The value of the central ^me #∗that maximizes ÅÇÈ^(#) is arctÆ an ^. (40) Since g is unknown, this op^mal value cannot be chose ¶n exactly. Compu^ng the deriva^ve of#∗ with respect to g , and using that arctan(¦) ≥ ?}¶V for ¦ ≥ 0 , wefind that #∗ is adecreasing func^on of |g| . Its maximal value is 1 / l(^) . Moreover, within the binary search approach described above, a1er 4 searches we have located the ground state with precision2^Ë, so we know that |g| ≤ 2^Ë. Hence, assuming we have |g| ≤ gs for some known ini^alguess gs > 0, we deduce the following bound?arctan Æ^? Æ^ ^ ≤ #∗ ≤ ^, (41) which depends only on known factors. In the following, we will set the central ^me equal to the lower bound #(^) = ?Æ^ arctan The total number of two-qubit gates in the circuit is ^ Hence it follows that the total run^me to answer one ques^on (35), assuming we knowa bound |g| ≤ gs, is with #(^) given in (42) and l(^) in (39). We recall that this total run^me is the number of two- qubit gates per circuit ^mes the number of circuits to run. In Figure 3, we plot the es^mate of the total run^me to get chemical precision 1mH with this binary search approach. Specifically, we plot the minimum with respect to the gate angle ^ of 10_^Æ,Æ^ (^) for g = gs = 10^K, with ^ = 10, ^ = 1 / 2, @ = 10, as a func^on of ^J. This valueof g corresponds to the last step of the binary search (when one already knows the loca^on of the ground state within 2 mH), and the factor 10 is a rough (over)es^mate of the cost of previous steps. The analy^cal value of the total number of CNOTs to run to get chemical precision at large ^J@ on noiseless hardware is with g = 10^K. In Sec^on 6 below, we will present noisy implementa^ons of this approach, andshow that a1er simple noise mi^ga^on the sign of the measured amplitude is indeed not affected by the noise. Figure 3 provides an es^mate of the minimal total run^me to get chemical precision with the 25 MJJ / 168922PCT1binary search approach, as a func^on of ^J, with ^ = 1 / 2, ^ = 10, for different values ofnoise `. The darker lines are obtained with @ = 10 and the lighter lines with @ = 20 (darkerand lighter lines are essen^ally superimposed for ` = 0). Total run^me is given in units ofnumber of shots ^mes number of two-qubit gates per shot, i.e. they are related to real run^mes by mul^plying by the average dura^on of a two-qubit gate and dividing by the average number of gates that can are executed in parallel. The values plo;ed are upper bounds. As an example, the values of ^Jfor chromium dimer in a STO-3G basis or FeMoCo with an ac^vespace of 50 states are around ^ KJ ≈ 10 .4.2.3. Arctanfit approach The approach we presented in Sec^on 4.2.2 above spares circuit depth compared to the “itera^ve" QPE by running a central ^me evolu^on with # just large enough so that the sign of the amplitude can be obtained. This allows one to answer the ques^on (35), namely whether the ground state energy is below or above ^. We argued that this approach is noise- resilient because noise, while certainly affec^ng the measured value of Å(#) would be less likely toflip the sign of Å(#), especially when # is large enough. An improvement can be made further if the noise is mild enough to be described by a simple depolarizing channel. Namely, we make the following assump^ons. Firstly, (i) we assume that the effect of the noise does not depend on g. This is a reasonable assump^on as g only enters as a phase in Å(#). If, for example, we compute Å(#) using a Hadamard test, the circuit depends on g only through one single-qubit rota^on. Secondly, (ii) we assume that the effect of the noise is just to mul^ply the measured Å(#) by some posi^ve factor ¢(#). This means that noise only acts as a global depolarizing channel. We jus^fied this assump^on in Sec^on 3.4.2 by poin^ng out that depolarising channels are the dominant sources of noise in many current hardware pla^orms and by referring to exis^ng noise mi^ga^on techniques that convert non-depolarizing noise into global depolarizing noise, like for example Probabilis^c Error Cancella^on [50, 51]. This provides a significant advantage over known techniques. Under these assump^ons, although noise will affect individual amplitudes Å(#) , it will not affect the value of ^ where Å(#) vanishes (which is the ground state energy). We will test numerically this fact in Sec^on 6.3 with noisy simula^ons. Specifically, let us assume we have an ini^al energy es^mate ^^È^^with anes^mated precision ^ > 0. We measure Å(#) for two different values of ^ = ^^È^^ ± ^ , butwith the same # , deno^ng űthe noisy measurement outcomes. Under these two assump^ons, we can extract ^!" = ^^È^^ + ?^ arctan ptan (46) The advantage of this formula is that it depends on a ra^o of measurements of Å(#) at same value of #, and so would not be highly sensi^ve to depolarizing noise as it would cancel out from numerator and denominator. Using this formula to refine the es^mate of the ground state energy is a way of u^lising informa^on in the amplitude of Å(#), which are, otherwise, thrown away in 26 MJJ / 168922PCT1 the binary search approach above (which only requires to know the sign of Å(#) ). We will present below in Sec^on 6 noisy numerical simula^ons of this approach and show that it performs well, showing that these assump^ons are realis^c. 4.2.4. Robbins-Monro approach The previous “arctanfit" approach relied on the observa^on that, in the presence of depolarizing noise, although the value of Å(#) is modified, the value of ^ where Å(#) vanishes is invariant, which is ^!". Wefinally present a third approach to compute ^!"that uses an algorithm tofind the zero of a func^on that we only know with sta^s^cal error. The Robbins-Monro algorithm is an algorithm forfinding the root of a func^on :(¦) when instead of having directly access to :(¦), one has only access to a random variable Ï(¦) thataverages to :(¦) , i.e. ^[Ï(¦)] = :(¦)

[0058] . Specifically, the Robbins-Monro algorithmrequires that (i) :(¦) is a non-decreasing func^on of ¦, (ii) :′(¦∗) > 0 where :(¦∗) = 0,and (iii) the values taken by the random variable Ï(¦) are uniformly bounded. In our case, thissetup corresponds to the variable ¦ = ^ the trial energy, to the func^on :(¦) =ℑ〈&|^^^(^^^)|&〉 at somefixed #, and with Ï(¦) the expecta^on value of one random circuit generated with the algorithm in Sec^on 3.3. The requirements of the Robbins-Monro algorithmare sa^sfied provided ^ is in the interval [^µ !" + ^^] . This algorithm takes asparameters an arbitrary sequence of steps Ð( > 0 such that Star^ng from an ini^al energy es^mate ^s, we then build a sequence of energy es^mates ^(as where Ï((^() is the expecta^on value of random circuit generated with the algorithm inSec^on 3.3 to measure equence of energies ^( converges when . →∞ to the root of the fu namely the ground state energy ^!". This approach shares with the previous arctanfit approach its noise resilience when the noise can be well approximated by a depolarizing channel, as it does not shi1 the value of ^ where Å(#) vanishes. It has the advantage of being a well-studied algorithm with op^mality results and improvements. However, the arctanfit approach above incorporates the known shape of the func^on Å(#) which might bring some addi^onal precision. We will present in Sec^on 6 a comparison of these different approaches using a depolarizing channel. 5. Resource es^mate In this Sec^on we report a resource es^mate for obtaining chemical precision with the TETRIS method on different molecules. 27 MJJ / 168922PCT1 5.1. Hamiltonian-related costs We start with a study of the required resources of the algorithm that pertain to the size of the molecular system studied. The number of qubits and the number of terms in the Hamiltonian ^ do not enter directly the cost of the algorithm. The only dependence of the number of two-qubit gates in the system size is through the interac^on norm ^J, which is the sum of the coefficients in the decomposi^on of the interac^on Hamiltonian ^Jinto Pauli strings, and through the factor ^, which is the average number of two-qubit gates required to implement a rota^on ^^^^^. Figures 4A and 4B show the dependence of these two parameters ^Jand ^ on the number of qubits, for various molecules and basis sets. The interac^on norm ^Jand the factor ^ are plo;ed as a func^on of the number of qubits (i.e. the number of spin orbitals). We can decompose these molecular Hamiltonians in terms of Pauli strings, a1er applying a Hartree-Fock op^miza^on of the molecular or spin orbitals. We call ^ the 1 -norm of the Hamiltonians obtained this way, and ^Jthe 1 -norm of the Hamiltonians without coun^ng single Pauli a terms, see Sec^on 3.3, and a1er removing the square par^cle number operator to minimize the norm as explained in Sec^on 3.5. We observe that the norm ^Jscales polynomially with thenumber of qubits ^, with afit ^ ^.Ó^J ≈ 0.007^ . When restricted to hydrogen chains, we observea behaviour ^ ^.?KJ ≈ 0.2^ . We note that it is known that if instead of doing the Hartree-Fockop^miza^on, we perform an orbital rota^on to minimize the norm, then we can bring this exponent below 2

[0032] . Moreover, we see that the norm ^Jof the interac^on Hamiltonian is o1en a few factors below the norm of the en^re Hamiltonian. The factor ^ is seen to be wellapproximated by ^ ≈ ^ / 2.In par^cular, the circles in Figure 4A show values for reduced norm ^Jand the crosses show total norn ^, and Figure 4B shows plots for factor ^ assuming a Hadamard test and Jordan- Wigner decomposi^on, as a func^on of the number of qubits ^. The molecules considered arehydrogen chains H ^( for . = 1, ... ,8 , hydrogen squares P (V ≡ H (V for . = 2,4 , hydrogencube Cub^Õ≡ H^Õ, N^, H^O, O^, LiH, BeH^, CO^, OK, BFK, CH^in a STO-3G basis, and H^, H^O, LiH, O^, CH^in a CCPVDZ basis (indicated by a ′ on the plot), all in their equilibrium configura^on. 5.2. Adiaba^c-path-related costs 5.2.1. Minimal adiaba^c ^me to reach chemical precision We now es^mate the behaviour of the required adiaba^c ^me evolu^on @ to reach chemical precision, as a func^on of the system size. We study two different paths that have different smoothness behaviour. Thefirst one is linear ~(l) = l , (49)for which we have ^ = 1 / 2, and the second one has a quadra^c behaviour both at l = 0 andl = 1~(l) = 2l^ − l^ , (50)for which we have = 7 / 15. We recall that the smoothness of the path influences the28 MJJ / 168922PCT1 asympto^c scaling of adiaba^c state prepara^on. In the le1 panel of Figure 5A, wefirst compare the energy ^(@) reached a1er an adiaba^c evolu^on during a ^me @ , with the ground state energy ^!"obtained with exact diagonaliza^on, for a hydrogen chain with 2 to 10 atoms in a STO-3G basis. Specifically, Figure 5A shows the difference between the ground state energy ^!"and the energy ^ of the stateadiaba^cally evolved up to ^me @, as a func^on of @, for hydrogen chains ^^, ^^, ^%, ^Ó, ^?s,and two different paths: ~(l) = l (darker lines) and ~(l) = 2l^ − l^ (lighter lines). Thehorizontal lines indicate precisions 10^Kand 10^^. These plots implement the exact adiaba^c evolu^on, without any Tro;er error or discre^za^on error. We observe a general decrease of^(@) − ^!" as ∼ 1 / @^ for the linear path, and around ∼ 1 / @^ for the quadra^c path. This isconsistent with the fact that smoother paths are associated with be;er asympto^c scaling. From these curves, we plot in the right panel of Figure 5B the smallest adiaba^c ^mes @Ç^(for which chemical precision 10^Kis reached in a hydrogen chain, as a func^on of the number of hydrogens, for linear and quadra^c paths. For both the linear and quadra^c paths, this ^me is seen to vary linearly with the number of hydrogen atoms in the chain. In the le1 panel of Figure 6A, we then consider a larger class of molecules for which we compute the minimal adiaba^c ^me @Ç^(required to reach accuracy 10^Kon the ground state energy, using the linear path (49) (circles) and quadra^c path (50) (squares), and plot it as a func^on of the number of qubits, for a subset of molecules appearing in Figure 4. For most of thesemolecules that are reachable with exact diagonaliza^on (^ 20 qubits), we observe that theadiaba^c ^me remains below 10, and can be bounded by ^ the number of qubits. This is in contrast with adiaba^c ^mes that were hypothesised to blow up exponen^ally with the number of qubits in certain previous works

[0013] . On the other hand the order of magnitude of most of the adiaba^c ^mes we found is in agreement with certain previous es^mates

[0015] . In the right panel of Figure 6B, we then consider non-equilibrium geometries by stretching the bond length of some of these molecules. Figure 6B is the same as the le1 panel, but as a func^on of the stretching defined as the ra^o of the bond length with the equilibrium value. These adiaba^c ^mes for stretched geometries are obtained with the standard paths we consider in this patent specifica^on, but they can be significantly decreased by modifying the path, see Sec^on 5.2.2 ‘Path modifica^on’ .For small stretching we observe only a mild varia^on of the minimal adiaba^c ^mes. However, we see that for larger stretching there can be a significant increase of the adiaba^c ^me with the par^cular path we study, star^ng from only single a Pauli strings. 5.2.2. Path modifica^on Besides the cases where the adiaba^c ^mes needed to reach chemical precision remain small, we also encountered a few cases not listed in Figure 6 where these ^mes become very large. For example, equilibrium hydrogen H^in a CCPVDZ basis set with 20 qubits in our setup requiresan adiaba^c ^me > 1000 to reach chemical precision. These ^mes are obviously very large andcould preclude using ASP to solve chemical systems. We also saw in the right panel of Figure 6 that adiaba^c ^mes tend to grow fast with the stretching. 29 MJJ / 168922PCT1 One advantage of ASP is the very large freedom one has in the choice of the adiaba^c path. Besides varying the speed along the path, one can also vary the ini^al Hamiltonian and include so-called counter-diaba^c terms [18, 59-65]. ASP can be slow for one such path and considerably faster for another path. Let us take the example of equilibrium H^in a CCPVDZ basis set, which is 20 qubits. As noted above, when star^ng from the Hamiltonian containing only single Pauli a matrices, the adiaba^c ^me required to reach chemical precision is larger than 1000. If now we take the ini^al Hamiltonian ^vas the one containing all Pauli strings in ^ with either exactly one or two a Pauli matrices (instead of just those with one a), then the adiaba^c ^me to reach chemical precision is reduced to around 4. A very simple modifica^on of the path can thus completely change the adiaba^c ^me required. Let us give another example with the non-equilibrium geometries at large stretching, for which ASP was observed to be slow when star^ng from the Hamiltonian with only single a terms. We propose instead the following adiaba^c path. Wefirst prepare the ground state of the equilibrium geometry by star^ng from the Hamiltonian with only the single a terms. As we saw, the minimal adiaba^c ^mes remain small for most of the molecules in that setup. Then, we slowlyincrease the stretching with ^me #(^) = #7(^^^°×(1 − ^r) + #^ Ø^(°× r, performing the Hartree-Fock op^miza^on of the Hamiltonian with that new value of stretching at each ^me step. We report in Figure 7 the energy precision obtained during this adiaba^c process when preparing the ground state of a molecule with stretching 3, star^ng from the ground state of the equilibriumgeometry (i.e. stretching 1 ), for H ^ and LiH. In par^cular, Figure 7 shows the differencebetween the ground state energy EGS at stretching value 3 and the energy E of the state adiaba^cally prepared evolved up to ^me T, as a func^on of T, for H2and LiH in a STO-3G basis. The adiaba^c path starts from the ground state with stretching 1 (i.e. equilibrium) and increases the stretching up to 3, during a total ^me T. The gray dashed line indicates chemical precision 10^(-3). We observe that this path yields much lower adiaba^c ^mes to reach chemical precision.Compared to the previous path, for H ^, this reduces the adiaba^c ^me from ≈ 300 to ≈ 15,while for LiH the reduc^on is from ≈ 800 to ≈ 100.We note that there exist s^ll several other ways of reducing the adiaba^c ^mes or the circuit depth [12, 66]. These examples show that theflexibility of ASP (in the choice of the path, the ini^al Hamiltonian, possible counter-diaba^c terms) can overcome unfavorable setups where adiaba^c ^mes needed naively appear to be very large. 5.3. Sampling costs The algorithm presented in Sec^on 3 is a randomized algorithm, in the sense that it requires to average over several different random circuits to obtain the desired quantum expecta^on value within the adiaba^cally evolved state. Compared to a determinis^c algorithm like a Tro;er decomposi^on where only one circuit is required, this seemingly comes with a sampling cost overhead. If these random circuits were simulated classically, there would be indeed an overhead in 30 MJJ / 168922PCT1 including mul^ple samples to compute the sta^s^cal average over the random circuits. However, on a quantum computer, expecta^on values are computed through averaging measurement outcomes over several realiza^ons of the same circuit, called shots. To obtain a precision ^ on the expecta^on value, one has to run in any case of order ^^^shots. In our case, because of the randomized algorithm we use, imprecision on the expecta^on values comes from two sources: thefinite number of random circuits, and thefinite number of shots per circuit. Given a total shot budget I, it is a priori non trivial how to best distribute these I shots among different random circuits, i.e., what is the op^mal number of shots # per circuits that minimizes the total varianceover the I / # random circuits. As shown in [1], this op^mal number is # = 1, namely one singleshot per circuit. In that case the noise coming from thefinite number of circuits and that coming from thefinite number of shots per circuit are indis^nguishable. Let us now es^mate the actual sta^s^cal variance associated with the random circuit drawing in the case where we do only one shot per circuit. We write an amplitude as with M?, M^ two independent random circuits drawn to generate Q(@), and M′? a randomcircuit drawn to generate . ^Ùand ^Úare the a;enua^on factors corresponding togenera^ng Q(@) and respec^vely. We will denote ^ = the total a;enua^onfactor. For −1 ≤ ¦ ≤ 1, we denote ℬ(¦) the Bernoulli random variable that takes value +1?}¶ ?^¶ with probability^ and −1 with probability^ . We denote then h the random variable This random variable is exactly the measurement outcome of one shot of a circuit that measuresthe imaginary part of 〈0|MN?M?»M^|0〉, mul^plied by ^^?. We thus have the mean value ^[h] = |0〉 . The number of circuits to include to converge to the mean isaround the variance ≈ VÐ`(h) withVÐ`(h) = ^[h^] − ^[h]^ . (53)In our case, since ℬ(¦)^ = 1, this variance is exactly Hence the sta^s^cal variance of the algorithm is bounded by the inverse square of the total a;enua^on factor VÐ` (55)This means we need at most ≈ 1 / ^^ shots to converge to precision ^(1), and ≈ 1 / (^^)^ toconverge to precision ^. Thus, in contrast with Tro;eriza^on (where the actual amplitude of Tro;er errors are a priori unknown), for any desired confidence interval, the required total cost (i.e. number of gates per sample ^mes number of samples) can be known in advance. Given a 31 MJJ / 168922PCT1 Tro;er step, there is no “guarantee” that the ^me evolu^on generated with Tro;er has converged to the exact ^me evolu^on. In contrast, with computer-implemented methods using the TETRIS algorithm, the only imprecision comes from thefinite number of shots as there is no discre^sa^on error. Therefore, given a number of shots and the variance associated to the measurements, one can affirm the probability that an exact value is within a par^cular range.In Figure 8, we evaluate numerically the sta^s^cal variance in the case # = 0 , measuring = 1 , the real part of the amplitude (which should be 1 ) instead of theimaginary part, with the randomized algorithm with 1 shot per circuit, as a func^on of T, for differ (ent gate angles τ(T), for a H4molecule at equilibrium in a STO-3G basis. Wefix the gate angle^ = r^w for . = 1,2,3,4 and vary @ . For this value of the gate angle the a;enua^on factorshould become independent of @ as @ grows, with value ^^? ∼ ^(. We observe indeed thisbehaviour. This shows that the randomized nature of the algorithm, provided the op^mal gateangle ^ ∼ ?r^w is chosen, does not bring any overhead compared to a determinis^c algorithmlike Tro;er. That is, the apparent addi^onal overhead of TETRIS in running dis^nct random circuits is actually non-existent, since any quantum algorithm has to be run several ^mes (normally with mul^ple different realiza^ons of the same circuit, called “shots”) and the measurement outcomes can be averaged to yield a result. By doing only one shot per random circuit, the random circuit sampling has zero addi^onal overhead compared to a determinis^c algorithm (if the number of random circuit samples is equivalent to a single circuit being run the same number of ^mes). 5.4. Discre^za^on costs of Tro.er Let us now evaluate the cost of implemen^ng ASP with a Tro;er decomposi^on, and compare with our proposal. To use the Tro;er algorithm to implement ASP, we have to approximate the adiaba^c evolu^on operator as with Δ^ a chosen ^me step, and ^(^) the ^me-dependent Hamiltonian of interest, evaluatedat ^me points 0, Δ^, 2Δ^, .... Then each of the term ^^â^^(^) at ^me l is itself wri;en as aproduct of exponen^als ≈ ∏= ^â^5(>? ^ ^(^)^^, with f((l) the corresponding coefficientof ^(l) in front of Pauli string x(. This implementa^on comes both with a discre^za^on error (thefirst approxima^on) and a Tro;er error (the second approxima^on), that both lead to a hea^ng, namely to an increase of the energy of the state prepared. In the le1 panel of Figure 9A, we plot the energy of the adiaba^c evolu^on as a func^on of thefinal ^me @, for differentfixed number of steps 1 / Δ^, for a H%hydrogen chain. Specifically, Figure 9A shows the difference between energy of the state prepared and ground state energy, as a func^on of adiaba^c ^me @ for different number of Tro;er steps with linear adiaba^c path, for the H%hydrogen chain. The hea^ng effect is clearly visible on the plot. At afixedfinite number of Tro;er steps, when increasing the total simula^on ^me @, there is inevitably an increase of the energy of the state prepared a1er some ^me @. This precludes reaching chemical precision if the number of Tro;ersteps is too low. In this par^cular example we see that we require ∼ 200 ^me steps to reachchemical precision. Since each Tro;er step requires to implement a rota^on ^^^^^for the 919 32 MJJ / 168922PCT1terms of that par^cular molecule, this yields ∼ 10æ rota^ons.When implemen^ng a Tro;er decomposi^on, one has to choose the order of the terms, and different orderings may display more or less pronounced discre^za^on error

[0067] . However, for these large number of Tro;er steps required to reach chemical precision, these varia^ons areobserved to be negligible. For example, for H % at ^me @ = 7 with 300 Tro;er steps, bypermu^ng randomly the terms in the Tro;er decomposi^on wefind energies in the range0.98 ± 0.03mH above the ground state. The curves in the le1 panel of Figure 9A would thus besimilar for different Tro;er orderings. In contrast, in our case with the algorithm of Sec^on 3, we do not have any discre^za^on error. In par^cular, we do not have the approximate decomposi^on of (56): ASP is implemented exactly. For the example of the H%hydrogen chain men^oned above, the algorithm we use requires ^rota^ons where ^ = 11.7 and @ = 7 , which is aro ^J und a factor 10 less than aTro;er implementa^on. Moreover, by mul^plying the gate angle ^ (which is a free parameterof the algorithm) by some factor ç > 0, one can always divide the depth of the circuit by ç,provided we can afford a shot overhead ∼ ^è. In the right panel of Figure 9B, we compare thenumber of rota^ons ^^^^^that have to be implemented to reach chemical accuracy using a Tro;er decomposi^on and with TETRIS, for several different molecules with a number of orbitals within reach of exact diagonaliza^on. Specifically, Figure 9B shows the minimal number of Pauli strings rota^ons ^^^^^(of length ≥2) to perform to reach chemical precision by adiaba^cally preparing the ground state with a linear path, using a Tro;er implementa^on (purple, circles) and a TETRIS implementa^on 1 / 2 T^2 μ_I^2 (cyan, squares), for different molecules. We observe a significant reduc^on of order 10^with TETRIS in the number of gates per circuit. This shows that the algorithm of Sec^on 3 is able to significantly reduce the cost of ASP. 5.5. Cost of other ^me evolu^on algorithms As men^oned in the introduc^on, there exist many other techniques to implement Hamiltonian evolu^on besides Tro;er. However, none of the known techniques are op^mal. Another randomized algorithm for Hamiltonian ^me evolu^on is the qDRIFT algorithm

[0068] . The algorithm has a gate count ^(^^^^ / ^) for ^me ^, 1-norm ^ and precision ^. However, it is known to have a large prefactor and to present important discre^za^on error, see e.g. [1]. There exist as well ^me evolu^on algorithms with more favorable scaling than Tro er for example LinearCombina^on of Unitaries (LCU)

[0020] . This algorithm has a gate coun éêëéêë ±), with -the number of terms in the Hamiltonian. Approxima^ng the prefactor and the log term by some constant É, the cost is thus É^^^@. This prefactor is known to be large. It follows that TETRIS has a be;er performance when ì @< É b^. (57) Using the scaling of ^ that we obtained, this is approximately 33 MJJ / 168922PCT1 @^ 100É^ . (58)This condi^on is largely sa^sfied for the molecules we studied in Figure 6 We note that by changing the orbital basis to minimize the 1-norm of the Hamiltonian, this condi^on could bebrought to @ ^ ^(^^)

[0032] . This shows the be;er prac^cality of TETRIS for quantum chemistryapplica^ons using current and expected near term quantum compu^ng capabili^es. 5.6. Scaling analysis To conclude this Sec^on, we perform a basic scaling analysis of the cost of ASP for chemistryapplica^ons, based on the previous resource es^mates. Using ^ = ^ / 2,preparing a state has a cost The cost of measuring the energy with precision 10^Kusing the different methods presented in Sec^on 4 is É= % ^ÇÈ°^^[È 10 ^^J . (60)For the hydrogen chain, we had the es^mates ^ ^.?KJ ≈ 0.2^ and adiaba^c ^mes @ ≈ ^ / 2.This yields the scaling This es^mate shows that for systems with less than ∼ 10K qubits, the measurement of theenergy is the most costly step, whereas for larger systems the adiaba^c prepara^on step is more costly. For more generic molecules, we had the es^mate ^ ^.ÓJ ≈ 0.007^ . The adiaba^c ^mesrequired are more difficult to es^mate, but on the molecules we tested at equilibrium we foundan upper bound @ ≈ ^. Hence in total, we get a rough upper bound es^mate of the total numberof CNOTs We again recall that these es^mates are obviously very dependent on the scaling of the adiaba^c ^mes @ required to reach chemical precision, that we es^mated for systems with up to 20 spin orbitals. 6. Noisy simula^ons 6.1. Setup We now present numerical simula^ons of the algorithm that take into account hardwareimperfec^ons. We compile every circuit into Pauli matrices, I, IN, ^, a rota^ons, and the two-qubit gate ^^îY]YV, which is the na^ve coupling gate in Quan^nuum’s ion-trap hardware

[0033] . We model hardware noise by a simple depolarizing channel a1er every ^^îY]YVgate. This kind 34 MJJ / 168922PCT1 of basic noise model gives a good approxima^on of many current pla^orms. 6.2. Symmetryfiltering The presence of symmetries in a system can be exploited to reduce the noise on the measurement outcomes of hardware or noisy simula^ons. If the unitary operator that a circuit implements commutes with some charge operator ­, then one can discard in the measurement outcomes all the shots where the value of ­ is modified compared to the ini^al value of ­. In our case, the Hamiltonians describing molecules always conserve the number of electrons with spin up or down separately. However, the use of the randomiza^on algorithm comes with asubtlety. Although the total Hamiltonians conserve par^cle number ­ = ∑=^>? a^, each term intheir Pauli string decomposi^on does not necessarily conserve par^cle number. A1er the Jordan-Wigner transforma^on, we obtain terms like h?h^ + i?i^ which commutes with par^clenumber a? + a^. However, in every random circuit generated by TETRIS, rota^ons with respectto h?h^and i?i^are performed separately, and these terms separately do not commute with +a^ and do not conserve par^cle number. When drawing randomly the gates, the circuitsthat we obtain thus do not individually conserve par^cle number, although on average they will. Even in the noiseless case, some shots will not conserve par^cle number. Discarding them would lead to a biased result. However, since we know these shots have to average to 0 in the result, one can impose that these non-par^cle-conserving shots contribute as zero in the average over the shots (which is different from discarding them, because the total number of shots that we use to average the results remains the same). In the presence of noise, shots that do not conserve par^cle number can come from either the fact that the circuit itself does not conserve par^cle number, or from the fact that an error has occurred. However, because non-par^cle-conserving shots can also occur in noiseless circuits, we cannot discard these shots in the noisy case, and one can only impose that they contribute as zero in the result. The consequence is that this symmetryfiltering does not remove noise, but only converts it into a global depolarizing channelthat maps thefinal density matrix Å into (1 with some noise level ^.There is however a less constraining symmetry that is sa^sfied at the level of each random circuit generated by the algorithm. This is the parity of par^cles with spin up or down. As this property is conserved individually in each circuit, no shots breaking the symmetry will be observed in the noiseless case, and one can systema^cally discard non-parity-conserving shots in hardware data or noisy simula^ons as one is certain that an error has occurred in those shots. This is the technique that we will use below when any symmetryfiltering is men^oned. 6.3. Noise-resilience In Sec^on 4.2 we proposed three different methods to extract the ground state energy from the measurement of the expecta^on value of the operator within the state prepared adiaba^cally. If the state prepared is sufficiently close to the ground state, this amplitude issin(#(^ − ^!")) with ^!" the desired ground state energy. We argued that these methods are35 MJJ / 168922PCT1noise-resilient because the main effect of hardware noise is to dampen the signal sin(#(^ −^!")) by mul^plying it by a posi^ve constant that is independent of ^. In this Sec^on, we start by checking this affirma^on with a very common noise model where a depolarizing channel on two qubits is applied a1er every two-qubit gate. In Figure 10 we show simula^ons for a stretched hydrogen H^molecule in a STO-3G basis with bond length 1.11Å, on 4 qubits, comparing the noisy simula^ons of with exact value, using depolarizing channels with amplitude 0.01 and the corresponding op^mal gate angle ^ as described in Sec^on 3.4.2. In par^cular, Figure 10 shows the imaginary part of the expecta^on value of in the state prepared with ASP, for a H^molecule with bond length 1.11Å,using an adiaba^c ^me @ = 12 with a linear path, as a func^on of g = ^ − ^!" expressed inmH, atfixed # = 20: exact curve sin(0.02g) (con^nuous black), raw noisy simula^ons (larger,thinner crosses) and noisy simula^ons a1er parityfiltering (smaller, thicker crosses). The noise is modelled by a depolarizing channel with amplitude 0.01 a1er every two-qubit gate. The dashedcurve is afit 0.46sin(0.02(g − 0.979)). We show both the raw data and the data obtained a1erparityfiltering as introduced in Sec^on 6.2. For these parameters there are around 200 two- qubit gates per circuit. Wefirst observe that the raw noisy data are hugely dampened comparedto the exact value. At g = ^ − ^!" = ±70mH, the amplitude is damped by a factor ≈ 0.131,which is very close to the rule of thumb 0.99^ss ≈ 0.134, namely thefidelity of every two-qubitgate raised to the number of two-qubit gates in the circuit. The parityfiltering is seen to have alarge effect and mul^plies the amplitude of the signal by a factor ≈ 3 − 4.Next, we observe that the main effect of the noise a1er parityfiltering is to dampen the curve.The value of ^ where the amplitude ℑ〈&|^^^(^^^)|&〉 vanishes (which is ^ = ^!") is barelyaffected by the noise. Byfibng the noisy data points a1er parityfiltering, wefind that the resultsfit well with a sinus. The amplitude is dampened by a factor 0.46 compared to exact, and the point where the curve vanishes is shi1ed by only 0.98mH. This confirms the noise- resilience of extrac^ng the ground state energy from the point where the amplitude vanishes. 6.4. Performance of the three measurement methods In Figure 11 we show simula^ons for a stretched hydrogen H^molecule in a STO-3G basis with bond length 1.11 Å, on 4 qubits. We show the results for the three different approaches to measure the energy presented in Sec^on 4, using the same adiaba^c state prepara^on withlinear path and adiaba^c ^me @ = 12. The energy of the state produced this way is 0.49mHabove the exact ground state energy. In the le1 panel of Figure 11A, we show the results of the binary search approach, where one determines whether the ground state energy is above or below a certain energy ^ by measuring the sign of ℑ〈&(@)|^^^(^^^)|&(@)〉. We consider two cases where ^ is at a distance −10mH and 5 mH from the ground state, and plot the measured value of the noisy amplitude as a func^on of # the central ^me, for a H ^ molecule with bondlength 1.11Å, using an adiaba^c ^me @ = 12 with a linear path, for two different values of36 MJJ / 168922PCT1g = ^ − ^!" , for a two-qubit gatefidelity 0.9982 , with 1000 samples and 100 shots persample. Con^nuous lines indicate values a1er parityfiltering, and dashed lines raw data. The black lines indicate the theore^cal noiseless values. We see that the raw unfiltered results aresystema^cally shi1ed because of noise. This occurs even for # = 0 , because the circuit s^llcontains the adiaba^c state prepara^on in this case. However, a1er using the parityfilter, we observe that up to the shot noise the results are compa^ble with the noiseless exact values. Inpar^cular, they allow one to dis^nguish posi^ve from nega^ve by going to central ^mes # ≈15 − 20. Nevertheless, performing the same calcula^on with a case where ^ is at a distance±1mH from the ground state would be challenging and would require to go to large values of # or much larger number of shots. In the right panel of Figure 11B, we then test the two other approaches, the arctanfit approach and the Robbins-Monro approach and show the precision on the es^mated energy for the two approaches as a func^on of the number of samples considered, in the same condi^ons as in the le1 plot of Figure 11A. We start from an ini^al energy es^mate 10mH below the exact ground state. For the arctanfit approach, we then measure |&(@)〉 at energies 30mHbelow and 10mH above the ground state (i.e., g = −10mH and ^ = 20mH), and plot formula(46) for # = 20 and ^^È^^ = ^!" − 10mH as a func^on of the number of samples. We see thatthe ground state es^mate improves immediately with just one hundred samples, and reaches chemical precision a1er around 1000 samples. As for the Robbins-Monro approach, we use theparameters Ð( = ?s(^.ñò star^ng at δ=-10mH and plot the energy iterates (48) as a func^on of thenumber of itera^ons. The black con^nuous line indicates 0.49mH the energy of the adiaba^c state prepared, and dashed lines indicate chemical precision 1mH. We see again a very fast increase in precision at the beginning. Values close to chemical precision are also reached a1er around 1000 samples. Op^mal choices of coefficients Ð(could improve further the results. Measuring mul^ple observables There exists a simple varia^on of the algorithm that allows one to measure mul^ple commu^ng observables when evolved at ^me ^, like if we were directly measuring the qubits, namely to measure 〈0...0|^^^^^^^^^^|0...0〉 , (1.1)for different observables ^. The TETRIS algorithm requires an ancilla ini^alized in |+〉, and weassume that the system qubits are ini^ally in |0...0〉. The ini^al density matrix is thus The states before the ⊗ denote the ancilla, and the states a1er the ⊗ denote the system qubits. When genera^ng one random circuit M according to the random process described previously, and applying it on the system qubits condi^oned on the ancilla to be 1, we prepare 37 MJJ / 168922PCT1 Å= ?|0〉〈0| ⊗ |0...0〉〈0...0| +?|1〉〈0| ⊗ M|0...0〉〈? ^^ 0...0| + ^ |0〉〈1| ⊗|0...0〉〈0...0|MN + ?|1〉〈1| ⊗ M|0...0〉〈0...0N^|M. (1.3) Let us now generate another random circuit M′, independently from M and according to the same random process, and apply it on the system qubits condi^oned on the ancilla to be 0. We get ^^. Averaging over the unitary M or M′ gives ^[M] = ^[M′] = ^^^^^ , (1.5)with ^ the a;enua^on factor. However, the average is unknown. We thus have,a1er averaging over different random circuits ^[Å] = ?|0〉〈0| ⊗ |áV^^^^ 0...0〉〈0...0| + ^ |1〉〈0| ⊗ ^ |0...0〉〈0...0|^ ^^^^|0...0〉〈0...0|^^^^^ + ?^ |1〉〈1| ⊗ Åó(¸ , (1.6)with Åó(¸: = ^[M′M|0...0〉〈0...0|M some unknown density matrix. Let us now measurethe observable h ⊗ ^, with h on the ancilla and ^ on the system qubits. In average we get This is exactly the expecta^on value of ^ at ^me ^ ^mes the a;enua^on factor ^^. Different observables ^ that commute can in principle be measured simultaneously. For example, if ^ are strings of a Pauli matrices, by measuring all the system qubits in the a basis, one gets access to all their expecta^on values simultaneously. This technique removes the specificity of the observables measured in TETRIS compared to using a Tro;er decomposi^on, and brings the algorithm as general as the Tro;er decomposi^on to compute the expecta^on values. Noise mi^ga^on technique: large gate angle extrapola^on In TETRIS, the gate angle ^ is a free parameter that does not modify the precision obtained on the result, in the sense that exact expecta^on values can be computed for all values of ^ . However, the gate angle ^ modifies the random unitaries M sampled and the a;enua^on factor ^. The angle of the rota^ons in M is ^, and there are in average ^^ / sin^ such rota^ons in each circuit M . Being able to tune a parameter to con^nuously change the number of gates in the circuit, but without changing the expecta^on values, is the ideal setup to perform Zero Noise Extrapola^on (ZNE). Namely, on an actual noisy hardware, by changing ^ one can measure the effect of noise on the result, and compensate for it. Specifically, let us run two values of ^, given by ^s and ®^s where ^s > 0 isfixed and38 MJJ / 168922PCT10 < ® < 1 , and let ö? and ö÷ be the expecta^on values obtained from the correspondingcircuits. We assume that these gate angles are small so that sin^ ≈ ^. Also, as usual with ZNE,we assume that at low noise level the effect of noise is linear in the number of gates, namely that ö÷ = Ð + ·-G(®) , (2.1)with Ð, · parameters and -G(®) the average number of gates in the circuits generated withgate angle ^ = ®^s. Other func^onal forms could be considered as well, such as an exponen^albehaviour ö÷ = Ð^^¹=ø(÷), and would not modify the discussion. The number of gates -G(®)is propor^onal to 1 / sin^ ≈ 1 / ^. Under these assump^ons, we obtain a mi^gated expecta^onvalue as öú]^÷úûù=^ = ?^÷ . (2.2)The TETRIS algorithm allows thus for a very specific implementa^on of ZNE through varia^on of the gate angle. Let us es^mate the standard devia^on obtained on the mi^gated result. The standarddevia^on üù=^ obtained on öù=^, ^ Considering that there is a certain total number of shots to distribute among and ü÷, wewrite ü^ ^ ^ ^? = #? / ¦ and ü÷ = #÷ / (1 − ¦) , with 0 < ¦ < 1 andvariance is obtained for ^ and then it takes the value 7. Conclusion In the work described in this patent specifica^on, the inventors inves^gated the relevance of adiaba^c state prepara^on for chemistry applica^ons, using a new algorithm (described in the Appendix below and in [1]) to implement ^me-dependent Hamiltonian evolu^on without Tro;er hea^ng. Firstly, the inventors evaluated the cost to adiaba^cally prepare the ground state of molecules within chemical precision 10^K, using a Tro;er decomposi^on and with the TETRIS algorithm.This determined that the method using TETRIS divides by a factor ∼ 10^ the number of two-qubit gates required to reach chemical precision compared to Tro;er, for several molecules within reach of exact diagonaliza^on on less than 20 qubits. Moreover, the inventors evaluated the minimal adiaba^c ^mes required to reach chemical precision and found in many cases that they either remain of order 1 or 10, or that modifica^ons of the adiaba^c path can dras^cally reduce them. For example, the adiaba^c ^mes required for a hydrogen chain scale only linearly 39 MJJ / 168922PCT1 with the number of atoms. The precision required in chemistry applica^ons is a technical challenge in terms of noise, number of measurements and run^me. Moreover, because TETRIS requires averaging of complex amplitudes and not their absolute values squared, one cannot use the efficient and op^mized techniques that have been developed to measure efficiently the energy of states prepared with VQE. The inventors proposed three different approaches to measure the energy of a state which is adiaba^cally prepared. The methods and systems implemen^ng the three approaches rely on the fact that when |&〉 is an eigenstate of ^ , the value of ^ where the imaginary part of vanishes (that is the eigenvalue of |&〉 on ^) is resilient to depolarizing noise channels. The inventors tested numerically these approaches on small molecules and showed that they are efficient. From these results, the inventors determined that adiaba^c state prepara^on is a viable approach to compute the ground state energy of molecules within chemical precision, and the determina^on of energy of an adiaba^cally prepared state is significantly enhanced by the described methods and systems. The methods are executable on advanced quantum computers such as Quan^nuum’s H2 trapped-ion quantum computer system, which provides the ability to implement two qubit gates on qubits other than nearest neighbour qubits. This avoidance of the limita^on to nearest neighbour connec^ons is significant, as it can greatly reduce the number of two qubit gates required to perform a computa^on such as applying rota^ons for ^me-evolu^on of a computa^onal state. Chemical Hamiltonians are typically long-range and highly-coupled when wri;en in terms of qubits, and so reliance on gates between nearest neighbour qubits would be a significant limita^on. On typical quantum computers, addi^onal gates imply addi^onal gate errors. The high-fidelity qubits and all-to-all connec^vity of Quan^nuum quantum computers allow efficient computa^on of ^me-evolved computa^onal states using the methods described herein, with gates being formed between remote qubits when required to reflect the interac^ons between orbi^ng electrodes in a real physical system. The described methods can achieve adiaba^c evolu^on with chemical accuracy for some physical systems (such as small molecules) without error mi^ga^on other than symmetryfiltering. Implementa^ons of the present inven^on on a quantum computer with high-fidelity gates enables ground state energy determina^ons that take account of stretched molecular geometry when compu^ng the energy of a ground state of a quantum system. Nevertheless, the absolute costs remain large, and many of today’s quantum computers will only be capable of applying the described techniques for a determina^on of energy for small molecules. At the algorithmic level, there are opportuni^es for further op^miza^on such as, for example, modifying the path or the speed along the path, modifying the orbital basis in order to minimize the 1-norm of the Hamiltonian, or star^ng from a different ini^al state with lower ini^al energy. Each of these op^misa^ons is expected to yield dras^c savings, reinforcing the conclusion that adiaba^c state prepara^on has significant advantages for solving chemistry problems on a near-fault-tolerant or fault-tolerant quantum computer. 40 MJJ / 168922PCT1 Appendix 1: Con^nuous Hamiltonian dynamics on digital quantum computers without discre^za^on error Introduc^on.— Hamiltonian dynamics is one of the most promising applica^on of current and near-term quantum computers [69, 70]. It is believed to be exponen^ally faster to perform on quantum computers andfinds mul^ple applica^ons, either directly or indirectly as subrou^nes of other quantum algorithms [6, 71-75]. Given a Hamiltonian ^ and a simula^on ^me ^, thereexist several ways to implement M(^) = ^^^^ on a digital quantum computer where only one ortwo-qubit gates can be used. The simplest approach uses product formulas such as Tro;eriza^onto decompose the dynamics into elementary gates M(^) ≈ ^^^^] ... ^^^^^ [70,76,77]. Thesemethods have been generalized and improved in many ways [19, 78-90]. In par^cular, the introduc^on of randomness [67, 91-95] such as in the qDRIFT algorithm [29, 68, 96-102] has par^cularly good theore^cal performance for non-sparse Hamiltonians ^, i.e. when the number of terms in the Hamiltonian grows faster than ^(^) where ^ is the number of qubits. These algorithms have in common that they only approximate the con^nuous ^me evolu^on with finite-depth circuits, and display discre^za^on errors (some^mes called Tro^er errors) that vanish only with infinitely many gates. This is par^cularly problema^c for adiaba^c state prepara^on (since Tro;er errors generically lead to an effec^ve hea^ng) [6, 45, 103] and in applica^ons like quantum chemistry [37, 104-108] where extreme precision is desired. More sophis^cated algorithms have be;er theore^cal scalings, such as linear combina^on of unitaries [19, 20, 21] (LCU), quantum signal processing [22-26] or quantum walks [109,110], but require a large resource overhead. As a consequence, at least in the near term, exact con^nuous ^me dynamics is considered to be the preroga^ve of non-gate-based, analog quantum simulators [111-115]. In this Appendix we introduce an algorithm to compute Hamiltonian dynamics of observables on digital quantum computers without discre^za^on error evenwith afinite number of gates, for any Hamiltonian ^ and unitary ℳ. The expecta^on value 〈ℳ(^)〉 is obtained as the average over random circuits drawn from a judiciously chosen distribu^on, mul^plied by a known factor. One can choose freely the angle ^ of the rota^ons entering the circuit, only modifying the mul^plica^ve factor and not the precision of the result. It allows one to decrease the gate count at the cost of an increased number of shots on a quantum computer, yielding a natural noise-mi^ga^on technique. The op^mal choice of gate angle yields a shot overhead of order ^(1) with a gate count per circuit ^(^^^^) where ^ is the 1-norm of the Hamiltonian, without any dependence on the precision desired, contrary to all previous algorithms. Since the gate count depends only on ^, the algorithm is par^cularly adapted to non-sparse Hamiltonians. It also straigh^orwardly generalizes to ^me-dependent Hamiltonians without any overhead. We demonstrate the algorithm with numerical simula^ons of the electronic structure of the stretched water molecule and of a 2D Ising model where it outperforms both Tro;er formulas as well as other randomised compila^on techniques. Simula^ng small-angle gates with large-angle gates.We first introduce the basic mechanism that underpins our algorithm. For a real number 0 ≤41 MJJ / 168922PCT1ý ≤ 1, an angle ^ and ^ any operator such that ^^ = z, we have the equality where This rela^on can be used to realize an effec^ve angle 0 < ^′ < ^ by applying ^^^H withprobability To see this, we consider a circuit that contains ^ many rota^ons ^^^»Hand that produces the wave func^on |&〉 . For a subset I of these rota^ons, we denote |&"〉 the wave func^on obtained with the same circuit but with dele^ng the rota^ons ^^^»Hcontained in I and replacing the remaining rota^ons ^^^»Hnot in I by ^^^H. Then, for any observable ℳ , by repeatedly applying (63), we have where the sum runs over all the possible subsets I, I′ of gates to delete in the original circuit.The expecta^on value that originally involves a circuit with angle ^′ > 0 can thus beexactly computed with a circuit involving only larger angles ^ > ^′ but in which some rota^onsare randomly removed. This however comes with an a;enua^on factor ^^^ < 1 whichincreases the total number of shots required to reach a given precision on 〈&|ℳ|&〉. The samereasoning applies to rota^ons with a nega^ve angle ^′ < 0 , with then ^ < ^′ . We note thatsimilar mechanisms have been used recently to implement arbitrary rota^on angles

[0116] . Con^nuous ^me limit. We now consider a Hamiltonian ^ that we write as a linear combina^on of - operators ^(that sa^sfy ^^( = z, i.e.^ = ∑=(>? f(^( , (67)which is always possible by wri^ng it as a sum of product of Pauli matrices. We would like tocompute the expecta^on value of a unitary ℳ within the wave func^on |&(^)〉 = ^^^^|&(0)〉.According to the Tro;er-Suzuki formula, we have: ^^^^ = lim (^^^»5]H] .. ^^»5^H^ ^ / ^»^»→sÎ. ^ ) . (68)For each . = , we choose 0 < ^ ^^»5( ≤ * / 2 and implement ^ ^H^ using thepreviously explained protocol with angle ^(. In the sequence of rota^ons appearing on the right- hand side of (68), instead of each rota^on ^^^»5^H^we thus apply ^^^^6G( (5^)H^with probability ý(given by The corresponding a;enua^on factor ^(is 42 MJJ / 168922PCT1 In the limit ^′ → 0, the probability of picking a rota^on ý( → 0 becomes a rare event and weconverge to a Poisson process. Namely, for each rota^on ^^^^6G( (5^)H^we obtain a sequenceof gate ^mes 0 < (Ë^) <... < ^( < ^ drawn from a Poisson process with rate |f(| / sin^(.For a given realiza^on @ of all these ^mes for different .’s, we denote the wave func^on obtained by applying the rota^ons ^^^^6G( (5^)H^ordered by ^me of occurrence ^((^), irrespec^ve of .. Averaging expecta^on values over a plurality of quantum circuitconfigura^ons for a transverse-field Ising Hamiltonian, with M ^^HH = ^ ac^ng on a set ofqubits provides an average that converges to the exact ^me-evolved expecta^on value 〈0|^^^^ℳ^^^^^|0〉, up to a known mul^plica^ve constant.Then, deno^ng ^ the expecta^on value with respect to two independent configura^ons @, @′,we have with the total a;enua^on This is the fundamental rela^on that defines our algorithm. Remarkably, the exact expecta^on value at ^me ^ under con^nuous ^me dynamics can be obtained without discre^za^on error,by only applying rota^ons with applica^on ^me ^( > 0 fixed. Moreover, the average numberof rota^ons in each circuit is which isfinite and only controlled by ^(and not by the precision wanted on the result. We note that the sta^s^cal variance in es^ma^ng 〈&(^)|ℳ|&(^)〉 is bounded by ^^^^?^. An important consequence of this a;enua^on factor ^^^^is thus to increase the number of shots required by a factor ^^^^^^to reach a given precision. Statement of the algorithm. We state here the algorithm deduced from the previous explana^ons. It takes as input a Hamiltonian ^ decomposed as in (67), a unitary observable ℳ, a ^me ^ and an ini^al state|&(0)〉, and outputs 〈&(^)|ℳ|&(^)〉. It takes 0 < ^( ≤ * / 2 as - parameters.1. Prepare the system in the ini^al state |&(0)〉. 2. For each . ∈ {1, ... , -}, draw an integer 4( from a Poisson distribu^on with|5 parameter^|^ 678|^^|. Then draw 4( real numbers ^(^)( , where 9 ∈ {1, ... , 4(}, uniformly atrandom between 0 and ^. These : ≡ ∑=(>? 4( numbers are collected into a set @.3. Deduce then the sequence A?, ... , AB ∈ {1, ... , -} such that the index .corresponding to the C-th smallest element of @ is AD. For 4 = 1, ... , : in this order, apply43 MJJ / 168922PCT1 the rota^on ^^^EF6G( (5EF)HEFon the system. 4. Apply the observable ℳ on the system. 5. Repeat steps (2) and (3) with ^(replaced by −^(. 6. Measure the overlap with the ini^al state and divide the result by ^^^^given in (72). On a quantum computer, this step requires to perform a Hadamard test. 7. Repeat steps (1) to (6) a number of ^mes and average the results. When measuring the overlap with the ini^al state, one has to choose a number of shots I to do per configura^on, i.e. how to distribute the resources between number of configura^ons and accuracy of each configura^on. If we neglect compila^on ^me, the op^mal number of shotscan be seen to be I = 1

[0117] .Measurement of the ^me evolu^on operator. The algorithm presented above can be adapted to compu^ng the amplitude 〈&(0)| It is o1en called Loschmidt amplitude in the condensed ma;er literature and can be used to determine ground state energies as well as microcanical expecta^on values [74, 118]. With thesame nota^ons as in Eq. (71), we have 〈&(0)|^^^^|&(0)〉 = ^[〈&(0)|&r〉] / §^^^^. One thus skipssteps (4) and (5) in the algorithm above, and replace the a;enua^on factor step (6). Op^mal angle on a perfect quantum computer. Formula (71) holds for any choice of gate angle ^(. Increasing the gate angle ^(decreases the number of rota^ons per circuit -^í^, but also decreases the a;enua^on factor ^^^^, and so increases the number of shots ^^^^^^that have to be performed to reach a given precision. Thetotal number of rota^ons _({^ ^^(}) = -^í^^^^^ to perform to get a precision of order ^(1) onthe result is thus a non-trivial func^on of the gate angles ^(, that reaches a minimum at some ^(∗that can be computed numerically. At large ^, wefind analy^cally that the op^mal angle is (74)for all . = 1, ... , - , where ^ = ∑=(>? |f(| is the 1 -norm of the Hamiltonian

[0117] . In thatlimit wefind ^ = ^^? / ^ wh ^ ^^^^ ich remainsfinite and -^í^ = 4^ ^ . Hence the gate count percircuit scales as ^(^^^^) . (75) The number of shots to perform to reach precision ^ scales as ^(^^^). But remarkably, theprecision ^ does not enter the gate count, which remainsfinite even when ^ → 0. This differsfrom usual algorithms such as Tro;er, qDRIFT or LCU, whose gate count we recall in Table 1. Table 1: Gate count of different algorithms for Hamiltonian dynamics. - refers to the number of terms in the Hamiltonian, ^ its 1-norm, ^ the simula^on ^me and ^ the 44 MJJ / 168922PCT1 precision reached on the result. Op^mal angle on a noisy quantum computer. The op^mal angle value (74) holds for a perfect, noiseless quantum computer. Let us explain how this formula is modified in presence of noise. For simplicity, we assume that the applica^on ofeach gate ^^^^6G( (5^)H^ comes with a signal a;enua^on ^^[^ with some (̀ > 0. This kind ofnoise model is known to be a roughfirst approxima^on of the effect of the noise on current hardware. Moreover, other types of noise can be converted into this global depolarizing noise through some noise mi^ga^on techniques [50, 51, 73]. In state-of-the-art devices typical two-qubit gatefideli^es are ^^[ ∼ 99.8% , in which case ` ∼ 2e − 3

[0033] . There are on average^|5^| 678^^such rota^ons per configura^on. The damping due to hardware imperfec^on is thus The total a;enua^on of the signal is ^^^^¢^^^. Assuming(̀small, the op^mal ^mes minimize the total gate count at large ^ are now

[0117] Time-dependent Hamiltonian. The algorithm can be generalized straigh^orwardly to dynamics with a ^me-dependent Hamiltonian ^(^). Decomposing with ^me-dependent coefficients f((^) , and deno^ng | the expecta^on value of operators ℳ can again be wri;en as an average over configura^ons(71). The configura^ons |& 〉 are produced by drawing ^mes 0 (Ë^)r <... < ^( < ^ froma Poisson process with ^me-dependent rate |f((#)| / sin^(, for each . = 1, ... , -, and applyingon the ini^al wave func^on each gate ^^^^6G( (5^(^))H^ ordered by ^me of occurrence # = The a;enua^on factor is then obtained by replacing ^|f^(| byqsd#|f((#)|. In prac^ce, we proceed as follows. We introduce ^ the func^on as well as ^^?( (l) its reciprocal, i.e. such that l . We replace step (2) of the^me-independent case by the following step: 1. For each . ∈ {1, ... , -}, draw an integer 4( from a Poisson distribu^on with^ parameter^(^) 678|^^|. Then draw 4( real numbers ^̃(^)( , where 9 ∈ {1, ... , 4(}, uniformly atrandom between 0 and ^ (^) ∑=( , and set These : ≡ (>? 4( numbers are collected into a set @. Then in step (3), the rota^on applied is ^^^EF6G( (5EF(^))HEF where # is the^me at which the rota^on is applied. For the backward propaga^on one has to reverse the order of the gates. The total a;enua^on ^^^^of step (6) is This algorithm produces the exact ^me evolu^on of the wave func^on under the ^me- 45 MJJ / 168922PCT1 dependent Hamiltonian ^(^), without errors arising from discre^zing the con^nuous func^on f((#) or Tro;er errors. Background Hamiltonian.The algorithm has the following simple improvement. Let us consider a subset of gates ^ ⊂{^(}(>?,..,= that all commute among themselves, i.e. [^, ^′] = 0 for all ^, ^′ ∈ ^, and denote^^°5¸G[í^(^ . If for these terms we take the limit of zero angle ^c → 0 in thealgorithm, the Poisson process almost always picks rota^ons of terms in ^ with very short angle. But since they commute this can be implemented with afinite product of ^^^Hwith ^ obtained by summing the different angles. Hence the algorithm is modified as follows. In step (2) we only draw ^mes corresponding to terms not in ^ , and use them to produce an ordered sequence of gates as in step (3). However, between each applica^on of these rota^ons ^^^^H^and ^^^FHF at ^mes #( < #Ë , we apply ^^Æ^^^^^EøW^^^^ with g# = #Ë − #( the differencebetween the two instants at which the rota^ons are applied. This reduces the a;enua^on factor ^^^^, which is now computed only including the terms not in ^. The Hamiltonian ^^°5¸G[í^(^thus plays the role of a “background" Hamiltonian that is always applied on the system, but ontop of which rota^ons ^^^^H^ with ^( ∉ ^ are applied.Numerical tests. We now show numerical simula^ons of our algorithm. We start with a noisy implementa^on on a non-sparse Hamiltonian as rou^nely encountered in quantum chemistry. Hamiltonians describing molecular electronic structure problems are typically decomposed in a basis set of fermionic orbitals, where they read with f’s canonical fermionic operators and - the number of orbitals considered. Wri;en in terms of Pauli gates through a Jordan-Wigner transforma^on, these molecular Hamiltonians involve thus ^(-^) terms with each ^(-) Pauli gates, incurring a prohibi^vely large gate count for any Tro;er implementa^on of the dynamics. We consider a H^O water molecule in a STO-6G basis with 14 orbitals, and use our algorithm to study the Loschmidt amplitude ℒ(^) = 〈^^|^^^^|^^〉 , (82)where |^^〉 denotes the Hartree-Fock ground state. We take a geometry with an angle H-O-H of 105íand with an elongated distance H-O of 2.2 Å, to have a significant difference between the exact ground state and |^^〉. On a noisy hardware, we propose to compute the ra^o (83)This ra^o is not sensi^ve to the depolarizing part of the hardware noise, but s^ll contains informa^on about the energy levels in the oscilla^on frequencies. In the le1 panel of Figure 13A we show a noisy simula^on of our algorithm that includes a depolarizing channel with probability 0.002 a1er each 2 -qubit gate, and observe very good agreement. In par^cular, Figure 13A shows a noisy simula^on of a computa^onal chemistry Hamiltonian, with gate angles (77), with 2000 circuits with 20 shots each. Next, we show numerical results for the ^me-dependent case of our algorithm. For this case, we 46 MJJ / 168922PCT1 consider the 2D square labce Ising model in a transversefield ℎ, whose Hamiltonian is with periodic boundary condi^ons. We implement the adiaba^c prepara^on of the ground stateat ℎ^ = 2.5, star^ng from ℎ = 0. For afinal ^me @^ we consider the ^me-dependent magne^cfield ℎ(^) = ℎ^sin(µµ^^ ^^sin(^r^) )

[0119] . In the right panel of Figure 13B we show thefinal energyper site obtained at ^ = @^, as a func^on of @^, together with the error bars for afixed numberof shots. Specifically Figure 13B shows an adiaba^c state prepara^on, noiseless simula^ons with10æ circuits and ^ = 0.04 (circles with a dashed line, the shade indica^ng one standarddevia^on), and exact value (lighter con^nuous line). We see the agreement with the exact values, as well as the broadening of the error bars as @^grows atfixed ^, as imposed by (72). Finally, we compare our algorithm tofirst-order Tro;er and qDRIFT in the case of the 2D squarelabce Ising model (84) with magne^cfield ℎ = 2, which is a typical model studied in condensedma;er. In Figure 12 we plot the real part of the Loschmidt amplitude 〈0|^^^^|0〉 in the 2D Isingmodel at ℎ = 2 in size 3 × 4, star^ng from the a = +1 product state |0〉, comparing Tro;er,qDRIFT and the algorithm described in this patent specifica^on, using a same gate angle ^ = 0.1for our algorithm and for qDRIFT, and with a same ^me step 0.1 for Tro;er. We take 10æshotsper point (which is necessary to have a precision < 0.01 with high confidence), and distributethem over 10Krandom circuits with 10^shots per circuit for our case and for qDRIFT. We see that our algorithm gives almost perfect results. In par^cular, it works considerably be;er than qDRIFT. We also see that it performs be;er than Tro;er. This shows the prac^cal use of our algorithm. Interpola^on between quantum ^me evolu^on and path integral representa^on.Some comments about the upper and lower limits of the angle ^ are in order. When ^ → 0, thedifferent rota^ons ^^^H^commute at order ^ and there are increasingly more rota^ons drawn from the Poisson process. With probability 1 in this limit, a single configura^on |&r〉 is equalto the exact ^me evolu^on ^^^^|0〉, and the a;enua^on factor (72) is equal to 1. When ^ =* / 2, we have ^^^H^ = 9^(, and so when ^( is a string of Pauli operators, the rota^ons can beimplemented with only one-qubit gates. Each configura^on |&r〉 remains thus a product state in the a basis and describes a classical trajectory in discrete ^me. Thus (71) becomes akin to a path integral representa^on, in the sense that a quantum expecta^on value is expressed as an averaging over exponen^ally many classical trajectories. These two limits provide thus an interes^ng interpreta^on of our algorithm as an interpola^onbetween a single quantum trajectory for ^ = 0, and exponen^ally many classical trajectories for^ = * / 2. Intermediate values of ^ near * / 2 allow one to decrease the entanglement presentin each configura^on | compared to the exact ^me-evolved state ^^^^|0〉, but at the cost of increasing the number of samples required through the a;enua^on factor (72). This couldfind applica^ons in the classical simula^on of quantum dynamics.Discussion.— We presented a quantum algorithm to compute con^nuous Hamiltonian 47 MJJ / 168922PCT1 dynamics that requires a circuit depth that is independent of the desired precision ^, contraryto many algorithms that require an infinite depth when ^ → 0. We show that by applyingrandomly gates on the ini^al state according to a well-chosen distribu^on, and mul^plying the result by a known amplifica^on factor, one achieves zero discre^za^on error while havingfinite- depth circuits. This amplifica^on factor however increases the number of shots required to reach a given precision. The algorithm is par^cularly adapted to non-sparse Hamiltonians as the gate count only depends on the 1-norm and not the number of terms. The algorithm and its mechanisms can be generalized in a number of ways. For example, decomposing a circuit into Clifford gates and @ -gates, one can implement each @ -gate by applying a ^^Yµ / ^-gate or no gate both with probability 1 / 2, which is a I-gate ^mes a globalphase. Hence, because of the a;enua^on factor ^ = cos(* / 8) for each replaced @-gate, t^hisenables one to sample classically from a circuit with ^ many @ -gates in a ^me ^^|éêë(þê6 ^)|^mul^plied by the simula^on ^me of the resul^ng Clifford circuits. The exponent matches the currently best known exponent

[0120] . Implementa^on Detail Op^mal number of shots per circuit. In the described computer-implemented method, an expecta^on value is expressed as an average over random circuits. Each of these random circuits is then implemented on a quantum computer with a certain number of shots. Expecta^on values are computed by averaging measurement outcomes over mul^ple shots – i.e. mul^ple realiza^ons of the same circuit. Let us denote I the number of shots per circuit and - / I the number of circuits. Atfixed - , we would like tofind the value of I that minimizes the variance on the result, taking into account both the shot noise and the sta^s^cal error due to the random circuits.Given a random circuit M drawn from the Poisson process, we denote −1 ≤ 4^ ≤ 1 the realpart of the complex amplitude that we measure with our algorithm. Given the real part of acomplex amplitude −1 ≤ 4 ≤ 1 , we denote ¦Ë the Bernoulli random variable that takesvalue +1 with probability?}Ë ^ and −1 with probability^. We denote ^ the expecta^on value with respect to the random circuit M, and 〈〉 the expecta^on value with respect to the shot noise. The exact value of the real part of the complex amplitude that we measure a1er averaging over random circuits and shots is 4= ^[〈¦Ë^〉] . (85)With I shots to evaluate 4^and - / I different circuits M , the es^mated value 4^^of 4 is where the Mc’s are independent circuits drawn from the Poisson process, and where the ?}Ë ¦ (^) ’s are independent Bernoulli rando^^Ë^^m variables with parameter^. Let us compute the variance of this quan^ty. We have 48 MJJ / 168922PCT1 where we used that (¦(^)Ë^^ )^ = 1. Then, averaging over the shots ?^V In the case where we have only one circuit M, i.e. I = -, we have ^ Ë^ − 〈4 ^^〉 = =, which is the usual variance of a Bernoulli random variable taking values ±1 with mean the general case, a1er averaging over the circuits we have Since # ≥ 0, we obtain that atfixed total number of shots -, doing only one shot I = 1 percircuit minimizes the total variance. Op^mal gate angles. When measuring an observable ℳ with our algorithm with gate angles ^(, in absence of noise, the total number of rota^ons to perform to reach a precision ^ on the result is _({^(}) / ^^, where _({^(}) = 2^ ∑= |5^|(>? 678^^ exp(4^ ∑= (>?|f(|tan(^( / 2)) . (91) Wri^ng   ∗^^_ = 0 at the op^mal gate angle ^( , wefind that it sa^sfies the equa^on The le1-hand side is bounded from below by 2^^. Hence at large ^, we necessarily have ^(→ 0 for all . to make the right-hand side go to ∞. In that limit we have The le1-hand side is independent of . , so the leading behaviour of ^∗( when ^ → ∞ isindependent of .. Deno^ng this leading behaviour ^∗, we thus have 2^^(^∗)^? = (^∗)^^, andso Let us now consider the noisy case where each rota^on ^^^^H^comes with an a;enua^onfactor ^^[^ . The damping due to noise per circuit is thus . The totalnumber of rota^ons to reach a precision ^ is thus _({^(}) / ^^, with now Wri^ng again   _ = 0 at the op^mal gate angl ∗^^ e ^( , wefind that it sa^sfies the equa^on 49 MJJ / 168922PCT1 Let us assume that when ^ → ∞ , the quan^ty ­ ≡ ?^[^þê6^∗^þê6V(^^∗ / ^) −678V^^∗does not go to 0.Then the same reasoning as above would apply and we would have ^∗( → 0. But then we wouldhave ­ → −∞, and so the right-hand side would become nega^ve, which cannot be. Hence wemust have ­ → 0. This yields an equa^on for when ^ → ∞ Assuming(̀small, this is Figure 14 is a schema^c diagram showing a poten^al implementa^on of a quantum compu^ng system 1000. The quantum compu^ng system 1000 comprises two components, a classical computer 1100 and a quantum computer 1200. The classical computer 1100 may comprise a known form of digital computer(s) including one or more processors for execu^ng program instruc^ons and memory for storing the program instruc^ons and data. The classical computer 1100 may comprise target data 1300, a controller 1400 (which may also be a component within the quantum computer 1200 instead) and a Hamiltonian 1500. The Hamiltonian 1500 represents a physical system and may comprise an evolu^on operator and one or more parameters. The controller 1400 may be used to manage various interopera^ons between the classical computer 1100 and the quantum computer 1200, for example, transferring a compiled quantum circuit (program) to the quantum computer 1200 for execu^on. The controller 1400 may also allow a user to specify sebngs for the program which are then applied during execu^on of the program. The target data 1300 may represent data rela^ng to the physical system that is being modelled. The quantum computer 1200 may comprise a controller 1400 (which may also be a component within the classical computer 1100 instead), quantum circuits 1600, and qubits (or qudits) 1700. The quantum circuits 1600 may be configured to interact directly with the hardware of the quantum computer, for example to create and manipulate the qubits 1700. The quantum circuit 1700 may be considered as somewhat analogous to a compiled program (low-level code) which has been adapted to run on the specific hardware implementa^on of the quantum computer, such as reflec^ng the number and connec^vity of the qubits and gates available on the quantum computer. More generally, it will be appreciated that the configura^on and architecture shown in Figure 14 is provided by way of illustra^on and not by way of limita^on and hence the approach described herein may be implemented on many different types of quantum compu^ng systems or pla^orms. Figure 15 shows a sequence of steps of an example of a method as disclosed herein for determining a physical property of a quantum state of a physical system. Opera^on 2000 comprises receiving or measuring input data. This input data may relate to the physical system that is to be modelled. Opera^on 2100 generates a computa^onal model (e.g., a Hamiltonian) including a prepared ini^al state and a ^me-evolving operator. This computa^onal model may 50 MJJ / 168922PCT1 represent the physical system that is to be modelled. The ini^al computa^onal state may represent a state of the physical system and the ^me evolu^on operator may perform an evolu^on over ^me of the prepared ini^al computa^onal state towards a target computa^onal state. Opera^on 2200 generates random quantum circuits. These random quantum circuits may comprise a sequence of rota^ons of one of the terms with afixed gate angle, drawn randomly and independent of each other with a rate given by a func^on of the gate angle. Opera^on 2300 executes the quantum circuits to generate ^me-evolu^on states. The quantum circuits may be executed on qubits or qudits of a quantum compu^ng system, which qubits or qudits are measured to output expecta^on values of the ^me-evolu^on operator. Opera^on 2400 computes an average of the expecta^on values over the generated quantum circuits. Opera^on 2500 rescales and outputs a determina^on of a physical property. The computed average may be rescaled by a factor that is a func^on of the gate angle. Figure 16 shows a sequence of steps of an example of a method as disclosed herein for determining the energy of a quantum state of a physical system. Opera^on 3000 receives input data. This input data may relate to the physical system that is to be modelled. Opera^on 3100 generates a computa^onal model (e.g., a Hamiltonian) including a prepared state and a ^me- evolu^on operator. This computa^onal model may represent the physical system that is to be modelled. The prepared computa^onal state may represent a state of the physical system and the ^me evolu^on operator may perform an evolu^on over ^me of the prepared computa^onal state. The operator may comprise a parameter that encodes the energy of the ^me-evolved computa^onal state and also comprise a parameter-dependent expecta^on value that is zero when the parameter is equal to the ground state energy of the state. Opera^on 3200 generates quantum circuits, op^onally using the random circuits of the method of Figure 15. These quantum circuits may correspond to the computa^onal model. Opera^on 3300 executes the quantum circuits to generate ^me-evolu^on states. The quantum circuits may be executed on qubits or qudits of a quantum compu^ng system to compute the effects of the ^me-evolu^on operator of the computa^onal model. If using the described method of Figure 15, Opera^on 3400 computes an average of the expecta^on values over a plurality of shots of the randomly generated quantum circuits (but the indirect measurement to determine a ground state energy of Figure 16 could be implemented without the averaging over random circuits of Figure 15). Opera^on 3500 determines a value of the parameter where the expecta^on value of the quantum operator is zero. Opera^on 3600 determines the energy corresponding to the determined parameter value. The energy may be the ground state energy of the ^me-evolved quantum state. 51 MJJ / 168922PCT1 Simula^ng sparse SYK model with a randomized algorithm on a trapped-ion quantum computer The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated quantum system that shows a strong signature of quantum chaos. Due to its chao^c nature, the simula^on of real-^me dynamics becomes quickly intractable by means of classical numerics, and thus, quantum simula^on is deemed to be an a;rac^ve alterna^ve. Nevertheless, quantum simula^ons of the SYK model on noisy quantum processors are severely limited by the complexity of its Hamiltonian. As a mi^ga^on for this problem, the real-^me dynamics of a sparsified version of the SYK model are simulated with 24 Majorana fermions on a trapped-ion quantum processor, taking advantage of the highfidelity of qubits based on trapped ions (such as provided in Quan^nuum’s quantum compu^ng systems). The approach is to adopt the above-described randomized quantum algorithm, TETRIS, and to develop an error mi^ga^on technique tailored to the algorithm. Leveraging the hardware’s high-fidelity quantum opera^ons and all-to-all connec^vity of the qubits, it is possible to successfully calculate the Loschmidt amplitude for sufficiently long ^mes so that its decay is observed. Based on the experimental and further numerical results, the future possibility of larger-scale simula^ons of the SYK model is assessed by es^ma^ng the required quantum resources. We present a scalable mirror-circuit benchmark based on the randomized SYK Hamiltonian and the TETRIS algorithm, which provides a be;er es^mate of the decay offidelity for local observables than standard mirror-circuits. The Sachdev-Ye-Kitaev (SYK) model describes a strongly-interac^ng quantum system that consists of - Majorana fermions with random ¢-body interac^ons [A1-A3]. Its connec^ons to condensed ma;er physics and non-Fermi liquids make it an a;rac^ve model to study strong electronic correla^ons and disorder in metals and cuprates [A4-A6]. The model has also gained significant a;en^on as it admits a two-dimensional gravita^onal holographic descrip^on in the infrared regime. In fact, it has been used as a paradigma^c model for “Quantum Gravity in the Lab” [A7-A11], a research effort to study quantum gravity phenomena through the holographic duality lenses, leading to new understandings (see e.g. [A6, A12, A13] for reviews). The SYK model exhibits a strong signature of quantum chaos by satura^ng a universal bound of quantum Lyapunov exponent at low temperatures [A14]. Despite many analy^cal studies that address the sta^c and dynamic proper^es of the SYK model under certain limits, further numerical inves^ga^ons are required to understand them for the other regimes, e.g.,finite -,finite-range interac^on, etc. Simula^on of real-^me dynamics is useful for addressing signatures of quantum chaos through the Loschmidt echo, out-of-^me-ordered correlators, spectral form factors, and other observables in the toolbox of condensed ma;er physics and quantum informa^on science.The real-^me dynamics of the SYK model are simulated with four-body interac^ons (¢ = 4) toexplore its feasibility on noisy quantum hardware. Digital quantum simula^ons of the SYK models on noisy quantum hardware have been severely limited due to their fully non-local interac^ons and the number of terms scaling as ^ (see [A4, A15, A16] for experimental works). We alleviate these issues by considering a sparsified version of the SYK model [A17-A23] and employing a recently developed randomized Hamiltonian simula^on algorithm, TETRIS (Time 52 MJJ / 168922PCT1 Evolu^on Through Random Independent Sampling) [A24]. This stochas^c algorithm enables Hamiltonian simula^on without discre^za^on error and is par^cularly suited for simula^ng the SYK model, leveraging the randomized nature of the SYK Hamiltonian. As a demonstra^ve example we focus on a single dynamical observable: the vacuum survival probability, or Loschmidt amplitude. We choose the Loschmidt amplitude [A25-A28] because it is a key observable to study quantum chaos in the SYK model due to its known connec^on [A27, A28] with out-of-^me-ordered correlators (OTOC) [A29-A34]. In our experiment, we evolve the system for ^me ^ under the sparse SYK dynamics employing TETRIS and calculate the survival probability of the all-zero state on a trapped-ion quantum processor, whose all-to-all qubit connec^vity facilitates the simula^on of non-local interac^on in the model. We carefully diagnose the error sources contribu^ng to the measurement outcomes. Moreover, we develop an error mi^ga^on technique that relies on proper^es of the randomized algorithm we use forcompu^ng the ^me evolu^on. Our protocol enables the successful simula^on of - = 24 sparseSYK model un^l the probability decay inherent to the model’s ideal dynamics is observed at %^ ∼1. Besides simula^ng quantum dynamics of the SYK model, we employ the TETRIS algorithm for benchmarking the noise effect of a quantum processor on a circuit used to es^mate a local observables. Mirror circuit benchmarks provide scalable benchmarking protocols, where one measures the survival probability of the ini^al state a1er an applica^on of a mirror circuit, that is, a unitary circuit followed by its inverse [A35-A37]. The performance of a quantum processor in running a circuit with a given number of gates is measured by the decay of probability from 1, the ideal value, with faster decays indica^ng stronger noise effects. In the present work, we consider a variant of the mirror circuit benchmarks, where we use two independently sampled TETRIS unitary circuits, M and M′, that agree only on average with the correct ^me evolu^on operator. Thus, the circuit M′NM provides a mirror-on-average circuit in the sense that it is propor^onal to the iden^ty operator only on average [A38]. We argue that this benchmark gives a be;er es^mate of noise on local observables than the standard mirror circuit benchmark when using this stochas^c algorithm for implemen^ng the ^me evolu^on. Finally, we conclude by es^ma^ng the required quantum resources by extrapola^ng our numerical and experimental results and address the feasibility of larger simula^ons of the SYK model. Setup.— The ¢ = 4 SYK model is a quantum mechanical model with - Majoranafermions &^ ({&^, &D} = 2g^D) described by the Hamiltonian [A2, A3],^ = ∑^^D^¸^× %^D¸×&^&D&¸&× , (E1)The couplings %^D¸×are independent random Gaussian variables with mean 0 and variance VVar[% K!'^D¸×] = The sparse SYK model is defined by [A17-A23]^ = ∑^^D^¸^× ý^D¸×%^D¸×&^&D&¸&× , (E2)53 MJJ / 168922PCT1 where ý^D¸× ∈ {0,1} are randomly sampled, being equal to 1 with probability ý , where ýcontrols the sparsity of the model. The average number of terms in the Hamiltonian (E2) isý = ^(ý-^). For gravita^onal physics to emerge at the infrared regime, one needs to retainan extensive number of terms on average by sebng the probability to for an --independent sparsity param Veter A [A17]. To maintain the extensive energy, the variance isrescaled as Var[%^D¸×] = K!'c=Õ . Upon taking the disorder average, wefind the 1-norm of theHamiltonian This Hamiltonian on - Majorana fermions can be expressed in terms of Pauli matrices throughthe Jordan-Wigner transforma^on into ^ = - / 2 qubits We write the corresponding spin Hamiltonian as with a Pauli string x(. The resul^ng SYK Hamiltonians are thus all-to-all coupled, with long Pauli strings of average length ^(^), and with random coefficients that can take both small and large values. These proper^es make these Hamiltonians difficult to simulate. Note that a different fermion encoding could be used, but local encodings [A39, A40] provide diminishing returns due to the non-local and all-to-all nature of the interac^ons. The ternary tree encoding [A41] could be an a;rac^ve possibility due to its favourable scaling, despite the need for addi^onal ancillas,and we will explore it later on. For the system size studied - = 24, we found that the Jordan-Wigner encoding was more efficient. The most widely used simula^on technique, Tro;eriza^on, scales par^cularly badly with ^ for these systems. For a recent work on higher-order product formulas focusing on the SYK model, see reference [A42]. The ^me evolu^on operator is approximated by # Tro;er steps using the first-order Tro;er decomposi^on, ^ Since the Pauli strings are of length ^(^) on average and the exponen^a^on of a Pauli string of length ^(^) requires ^(^) two-qubit (TQ) gates, the cost to implement one single Tro;er step is ^(ý^æ) TQ gates. Equa^on (E6) comes with an discre^za^on (Tro;er) error ^(^^ý^^ / #) . The Tro;er error arises from non-vanishing commutators of the form 54 MJJ / 168922PCT1[&^&D&¸&× , &^»&D»&¸»&×»]. For this to befinite, two four-body terms must share at least one ofthe fermion indices. There are ^(-)) such combina^ons of (9CA*) and (9′C′A′*′). Moreover, each four-body term labelled by (9CA*) exists in a single disorder realiza^on of the Hamiltonian (E2) with probability ý . Therefore, the number of non-vanishing commutators scales as^(ý^-)) , resul^ng in the Tro;er error ^ ^ ) ^ ^ × ^(^ ý - / #) = ^(^ ý- / #) .Thus, the total TQ gate cost is ^(ý^^+^^ / ^) to achieve the Tro;er error ^. Sebng ý ∼ ^^K, asrequired to reproduce the proper^es of the dense SYK model, wefind a gate count ^(^K^^ / ^). Various randomized algorithms for Hamiltonian simula^on provide favourable quantum resources compared to the Tro;eriza^on in a number of cases [A43-A50]. As we will see below, the randomized TETRIS algorithm [A24] involves ^^^^rota^ons of Pauli strings, where ^ is the 1-norm of the Hamiltonian. Most notably, the gate cost of each circuit is independent of the required error ^ . Since each rota^on requires ^(^) TQ gates, this yields a TQ gate count ^(ý^%^^). The scaling is strictly be;er than the Tro;eriza^on scaling for any sparsity parameterý ∼ ^^÷ with ® < 3. For ý ∼ ^^K, TETRIS achieves no discre^za^on error at the same cost asthe Tro;eriza^on. Furthermore, no ma;er how small the evolu^on ^me ^ is, the Tro;eriza^on needs to apply at least one Tro;er step, which costs ^(ý^æ). On the other hand, the TETRIS gate cost decreases as ^^, and thus, becomes cheaper for short-^me evolu^on. This will be numerically verified later in Fig. 21. To give some precise numbers beyond this scaling analysis,for - = 24 Majorana fermions, we get an average number of two-qubit gates around 1022per Tro;er step, before circuit op^miza^on. For the TETRIS algorithm at op^mal gate angle, we get an average number of two-qubit gates around 3100^^for simula^on ^me ^ before circuit op^miza^on.The TETRIS algorithm takes as a parameter a gate angle 0 < ^ < * / 2, and generates randomunitary operators M that average to ^[M] ^^^^ = ^^ , (E7)where ^^denotes the sta^s^cal average over random samples of M, and ^ is ^= ^^^^^^8(^ / ^) , (E8)with the 1 -norm ^ = ∑( |f(| of the Hamiltonian (E5). The random unitary operator M isexpressed as a product M = ^^^^X] ... ^^^^X^ , where the mul^plica^on by ^^^^^ are randomevents that follow independent Poisson processes with rates |f(| / sin^ during an evolu^on ^me ^. As a consequence, the average number of rota^on gates per sampled unitary operator M is ^^ / sin^. The a;enua^on factor ^ increases the number of shots to reach a given precision by afactor 1 / ^^ choice of ^ that minimizes the total number of gatesacross different shots is ^ ≈ 1 / (^^) for large ^^. This yields an average number of rota^ons^^^^. Note that for the SYK Hamiltonian, the long Pauli strings that will be exponen^ated result mostly in two-qubit gates with maximal angle, coming from the decomposi^on of Pauli gadgets. 55 MJJ / 168922PCT1 Noise mi^ga^on: Echo verifica^on.— The implementa^on of TETRIS involves an ancillary qubit to calculate the average of unitary operators M. We discuss an error mi^ga^on scheme that makes use of the measurement outcomes on the system register [A51-A54]. Let us consider a density matrix Å on the ancilla and the system qubits, that we ini^alize in |+〉〈+| for theancilla, and |0〉〈0|: = |0...0〉〈0...0| for the system qubits. A1er applica^on of a unitaryoperator M condi^oned on the ancilla being |1〉, we have where the registers before and a1er ⊗ denote the ancilla and the system qubits respec^vely. Averaging over M, we get One notes that the average of M|0〉〈0|MNis a priori unknown since it is quadra^c in M . Measuring h on the ancilla and ^, an arbitrary observable, on the system qubits, we have Tr[Å (h ⊗ ^)] ^^^^$È = ^ℜ[〈0|^^ |0〉] . (E11)The real part of the Loschmidt amplitude can thus be obtained by sebng ^ = z , i.e., onlymeasuring the ancilla ℜ[〈0|^^^^|0〉] = ^^?Tr[Å^$È(h ⊗ z)] . (E12)However, it can also be equivalently obtained by sebng ^ = |0〉〈0|, the projector onto the ini^alstate, ℜ[〈0|^^^^|0〉] = ^^?Tr[Å^$È(h ⊗ |0〉〈0|)] . (E13)When measuring the system qubits in the a basis, the difference between the two cases aboveis that when compu^ng h ⊗ z, the measurement outcome of the system qubits is not taken intoaccount, whereas when compu^ng h ⊗ |0〉〈0| , we count zero whenever the measurementoutcome of the system qubits is not 0...0. This is different from discarding the correspondingshots because the overall number of shots remains the same. Similarly, projec^ng the systemqubits onto any product state in the a basis that is different from |0...0〉 , i.e., taking ^ = ..9b| has expecta^on value 0 whenever 9 0...0. It follows that coun^ng56 MJJ / 168922PCT1the shots where the system qubits are not 0...0 does not modify the expecta^on values, butincreases the shot noise. On a noiseless quantum computer, it is thus always more efficient tomeasure h ⊗ |0〉〈0|. This noise mi^ga^on technique is called echo verifica^on [A51, A54]. InFig. 17A, we show the variance over circuits, for different numbers of shots per circuit anddifferent numbers of gates, and for ^ = z , ^ = |0〉〈0| . We observe that ^ = |0〉〈0| hasgenerally smaller variance than ^ = z, as expected. As for the influence of the number of shotsper circuit, it is known that doing 1 shot per random TETRIS circuit minimizes the variance at fixed total number of shots, when neglec^ng compila^on cost [A24]. We observe that in prac^ce, doing 10 shots per circuit reduces the standard devia^on by around√10, which is thus close to op^mal. On the contrary, doing 10^shots per random circuit reduces the standard devia^on by much less than √10^which is far from the op^mal case.At early ^mes before the Loschmidt amplitude decays, |0...0〉 has the dominant amplitude inan evolving state. On a noisy quantum computer, some bitflips can occur to change the state|0...0〉 to a state of a small Hamming weight. Coun^ng 0 for these states will thus a;enuatethe signal. Since fermionic parity is conserved in the SYK model, shots with an odd number of bit flips can be a;ributed to noise with certainty. For a state with a small Hamming weight, noise is more likely to increase the weight than to decrease it. Therefore, the majority of the shots witha single 1 are expected to come from a bit-flip on top of 0...0 than on top of a bitstring withHamming weight 2. Taking these shots into account should thus improve the es^mate of h ⊗|0〉〈0| on a noisy quantum computer. To account for this, we propose a variant of the echo verifica^on by measuring the following observable ...01D 0...0〉〈0...01D 0...0| , (E14)where the single 1 is at posi^on C in each bitstring. Figure 17A discloses standard devia^on of measurements of real part of the Loschmidt amplitudeat %^ = 0.5, as a func^on of number of CNOTs per circuit (without any circuit op^miza^on), fordifferent number of shots per circuits (different shades of green) and measurement operator^ = z (solid lines) and ^ = |0〉〈0| (dashed lines). The circuits are sampled both from TETRISand from different disorder realiza^ons of the SYK Hamiltonian. Figure 17B discloses the real partof Loschmidt amplitude as a func^on of 1 / ® for gate angle ^ = ®^s with ^s = 1 / (^^) , for%^ = 0.5, simulated with a depolarizing noise channel of amplitude 0.001 a1er every TQ gate,for different operators ^. The error bars correspond to 10Kcircuits with 10 shots each. The dashed lines are linearfits and the do;ed line is an exponen^alfit. The black line shows the exact value. Noise mi^ga^on: Large Gate Angle Extrapola^on (LGAE).— In TETRIS, the gate angle ^ is a free parameter that does not modify the precision obtained on the result: the real part of the Loschmidt amplitude can be obtained with Eq. (E13) for all ^ . Yet, the gate angle ^ has a significant impact on the unitaries M sampled in Eq. (E7) and on the a;enua^on factor ^ in Eq. (E8). We leverage this freedom to introduce another error mi^ga^on scheme. The angle of the 57 MJJ / 168922PCT1 rota^ons in a TETRIS circuit M is ^, and there are on average ^^ / sin^ such rota^ons in each circuit M. Being able to tune a parameter to con^nuously change the number of gates in the circuit, but without changing the expecta^on values, is the ideal setup to perform Zero Noise Extrapola^on (ZNE) [A55, A56]. Namely, on an actual noisy hardware, by changing ^ one can measure the effect of noise on the result, and compensate for it. We refer to this variant of ZNE as Large Gate Angle Extrapola^on (LGAE), in the sense that extrapola^ng noise to zero in the ZNE is performed here by increasing the gate angle.LGAE proceeds as follows. We select two values of ^, afixed ^s > 0 and a rescaled ^÷ = ®^s,where 0 < ® < 1. Then, we run TETRIS with both angles and let ös and ö÷ be the expecta^onvalues obtained from the corresponding circuits. Let us assume for simplicity that these gateangles are sufficiently small so that sin^ ≈ ^. Also, as usual with ZNE, we assume that, at lownoise level, the effect of noise is linear in the number of gates, ö÷ = Ð + ·-ë(®) , (E15)with the parameters Ð, · and the average number of gates -ë(®) in the circuits generatedwith gate angle ^÷ . This number of gates is propor^onal to 1 / sin^÷ ≈ 1 / ^÷ . Under theseassump^ons, we obtain a mi^gated expecta^on value as ö,^Ù^ = ú^^÷úû?^÷ . (E16)The standard devia^on ü,^Ù^obtained on ö,^Ù^is where üs and ü÷ are the standard devia^ons of ös, ö÷, respec^vely. Considering that there isa certain total number of shots to distribute among ü ^ ^s and ü÷ , we write üs = #s / ¦ andü^÷ = #^÷ / (1 − ¦), with 0 < ¦ < 1 and #s, #÷ numbers. The minimal variance is obtained for^ and then it takes the value ü^^}÷^û,^Ù^ = ?^÷ . (E19)Similarly to the conven^onal ZNE, instead of the linearfit with the number of gates, we may employ an exponen^alfit. The assump^on is now that the effect of noise is of the form ö¹= (÷)÷ = Ð^ - . (E20)In that case the exponen^al mi^ga^on gives 58 MJJ / 168922PCT1 öéêëú ^÷é¶c^b^Ù^ = exp ^ êëúû?^÷ . (E21)There is no closed-form expression for the standard devia^on of this es^mate in terms of üs, ü÷in general, but assuming these are small, we can write using Gaussian propaga^on of uncertainty In Fig. 17B, we show the measured real part of the Loschmidt amplitude for different values ofgate angles, when performing noisy numerical simula^ons of the SYK model for - = 24fermions, and for the three operators ^ = z, |0〉〈0|, |0〉〈0|Ç^^ discussed in the previous sec^on.When running noiseless simula^ons, all the points would match the exact value, independentlyof the gate angle. However, in noisy simula^ons, we see that the signal measured for ^ =|0〉〈0|, |0〉〈0|Ç^^ depends strongly on the gate angle, and decays when the gate angle decreases.The signal for ^ = |0〉〈0|Ç^^ is approximately linear, while that for ^ = |0〉〈0| presents acurvature. The signal for ^ = z is much less sensi^ve to gate angle, and is even slightly largerthan the exact value due to the noise effect that we will discuss later (see e.g. Fig. 20). Because of the peculiar non-monotonous effect of noise here, the noise mi^ga^on works less well. Theerror bars show that ^ = z displays more shot noise, requiring around twice as many shots toget the same precision as for ^ = |0〉〈0|, |0〉〈0|Ç^^.Hardware results.— We now present a hardware implementa^on of our setup on Quan^nuum System Model H1 [A57]. This is a trapped-ion quantum compu^ng device hos^ng20 all-to-all coupled qubits. Wefix the number of Majorana fermions to - = 24, correspondingto ^ = 12 qubits, and add one ancilla qubit, using thus 13 qubits in total. The sparsityparameter is A = 2.3, below which the ramp, a universal behaviour predicted by random matrixtheory, disappears in the spectral form factor, indica^ng the loss of spectral rigidity [A23].Therefore, with A = 2.3 we choose the most sparse SYK that could s^ll be compa^ble withchao^c proper^es related to its holographic dual gravity model. We intend this experiment to provide a lower bound on the number of gates needed to simulate a SYK model with chao^c proper^es. On the hardware, we run 6 shots per circuit, each circuit being randomly sampled among different TETRIS unitary operators M and among different disorder realiza^ons of the sparse SYK Hamiltonian ^ (E2). We use the ability of the hardware to perform mid-circuit measurement and qubit reset to “s^tch” mul^ple circuits in a same run, in order to minimize the overhead cost in running several circuits with few shots each. We consider two different gateangles, ^s = 1.5 / (^^), where ^ is the 1-norm of the sampled Hamiltonian ^, that we refer toas “shallow circuits" and one equal to ^÷ = ®^s with ® = 1 / 3 , that we refer to as “deepcircuits". The deep circuits contain, on average, around three ^mes more gates than the shallow circuits. Taking a large difference of number of gates in the two circuits used to perform ZNE reduces the shot noise amplifica^on by the extrapola^on [A58]. This allows us to apply the LGAE on the shallow and deep circuits and to study the efficiency of the three different measurementstrategies, where ^ = z, |0〉〈0|, |0〉〈0|Ç^^. All hardware results are summarized in Fig. 18.Figure 18 discloses hardware results from the Quan^nuum H1-1 ion-trap quantum computer. 59 MJJ / 168922PCT1 Real part of the Loschmidt amplitude, averaged over random realiza^ons of the SYK Hamiltonianfor - = 24 fermions, as a func^on of ^me %^, for shallow and deep circuits, together with theLGAE mi^ga^on technique, for the three tested measurement operators ^. The black line shows the exact value.We observe that for ^ = |0〉〈0|Ç^^, the LGAE mi^ga^on recovers the exact values within justabout one standard devia^on. In contrast, for ^ = |0〉〈0| the mi^ga^on leads to anunderes^ma^on of the exact result. We a;ribute it to the fact that, because this observable is noisier, we are beyond the regime where noise effects are linear in the number of gates, as we already saw in Fig.17B. Figure 19 discloses hardware results a1er exponen^al LGAE, in the same sebng as in Fig.18. Themi^gated results for %^ = 0.8 have very large variance and are outside of the plo;ed window.In Fig.19 we plot the mi^gated results using the exponen^al form of LGAE. As expected, the errorbars are significantly larger. Here, even the case ^ = |0〉〈0| correctly recovers the exact resultwithin error bars, although the error bars are too large to be really conclusive for %^ = 0.65 and%^ = 0.8. For ^ = z with the linear extrapola^on in Fig.18, the results display significantly largeerror bars, with both shallow and deep circuits giving comparable results. In that case the efficiency of the noise mi^ga^on is also less conclusive due again to the large error bars. We also observe that the signal is typically larger than the exact values. This upward bias due to noise is not standard because noise typically biases the results towards vanishing signal, except for non- unital channels such as leakage error. We next study theore^cally the origin of this effect. Effect of noise.— We now discuss the effect of noise on the expecta^on valueTr[Å^$È(h ⊗ ^)] . We use 〈h ⊗ ^〉 to denote the noisy expecta^on values, whileTr[Å^$È(h ⊗ ^)] denotes the noiseless expecta^on value.We analyze the dominant source of error occurring on the system qubits. To get tractable expressions, we model the noise effects on the system qubits with a global depolarizing channel 3 that occurs with rate ¢. If the channel 3 is applied at ^me ^?, the density matrix (E10) is turned into with a;enua^on factor up to ^me ^?. Further evolving the state un^l^me ^^, wefind 60 MJJ / 168922PCT1by averaging over the TETRIS circuits, with ^^ = ^^^V^^^8(^ / ^). Hence, if . errors occurs at ^mes0 < ^? <... < ^( < ^, we measure the contribu^on at ^me ^ The probability of having . errors is ^d^(!^ . The measured value constrained to having .errors is expressed by the integral of (E25) over 0 < ^? <... < ^( < ^ with a normaliza^onfactor .! / ^(. Collec^ng the contribu^ons with different numbers of errors, it can be concisely wri;en as by introducing ^(^) sa^sfying the integral equa^on ^(^) = 〈0|^^^^|0〉 + ¢ q^s d# 4^[ÈUà(^^Þ)] ^^^(#) . (E27) For example at order ^((¢^)^), this is At a low error rate ¢ and for ^ = |0〉〈0|, we have Because of the factor 1 / 2=, the parenthesis is very likely to be nega^ve at small ^. Hence thesignal is decreased. On the other hand, when ^ = z, we have61 MJJ / 168922PCT1 〈h ⊗ 〉 = ^ℜ〈0 ^^^^ 0〉30) 4^ ÈUà^]] Now, the quan^ty ^ is of order 1 at small ^?. Hence, the parenthesis can be posi^ve or nega^ve, implying noise can amplify the signal, counter-intui^vely. Since Tr[^^^^] and 〈0|^^^^|0〉 are concave func^ons of ^ at small ^, we can expect indeed this parenthesis to be posi^ve. Figure 20 discloses the real part of Loschmidt amplitude as a func^on of ^me %^ , showing hardware results (bullets) and curves obtained from theore^cal noise model (dashed lines), forshallow and deep circuits, and for measurement operator ^ = |0〉〈0|, z. The single parameterentering the theore^cal model isfi;ed with the hardware data obtained for shallow circuits and^ = |0〉〈0| (lightgreen). The three other dashed curves are deduced from it. The black lineindicates the noiseless value. To compare this theore^cal noise model to hardware results, we proceed as follows. Recall that ¢ is the error rate per unit of physical ^me ^, not per gate. In an actual implementa^on of the algorithm, the error rate ¢ depends on the chosen gate angle, because the number of gates per unit of physical ^me depends on the gate angle. In the experiments, since the gate angle is set to be inversely propor^onal to physical ^me ^, the error rate ¢ is propor^onal to thefinal physical ^me ^ for each experiment. Note in the formulas above, ¢ stays constant in ^me, because the gate angle is constant throughout a single ^me evolu^on. To determine the propor^onality factor5 between error rate ¢ andfinal physical ^me ^, wefit the formula for 〈h ⊗ ^〉 in (E26) for^ = |0〉〈0|, only on shallow circuits. Since the error rate must be propor^onal to the number ofgates per unit of physical ^me, the error rate for deep circuits must be three ^mes larger.Thefit yields the value 5 = 2.46%^. This means that, for %^ = 0.5 for example, the error rate ¢in the shallow circuits is 5^ = 1.23%, and so we have on average ¢^ = 0.615 errors during theevolu^on for ^me ^. Since there are on average 275 TQ gates per shallow circuit for %^ = 0.5,this gives an average number of errors per TQ gate equal to 0.0022. The component benchmark values give an error rate per two-qubit gate around 10^K[A57]. However, this value 0.0022 obtained from the hardware also takes into account other sources of error like memory error, it is thus expected that it is larger than the component benchmark value. We plot in Fig. 20 the comparison of the theore^cal noise model obtained with thisfi;ed parameter, for the threeother sebngs, namely shallow circuits with ^ = z, and deep circuits for both ^ = |0〉〈0| and^ = z . We observe a good agreement for all the setups. In par^cular, the model correctlycaptures the upward bias due to noise of hardware results when ^ = z.Figure 21A discloses the real part of Loschmidt amplitude as a func^on of ^me %^, for differentsystem sizes - = 8,12, ... ,32 (from right to le1), with green indica^ng that TETRIS at op^malangle is cheaper than Tro;er, orange that one Tro;er step is cheaper, and purple that two Tro;er 62 MJJ / 168922PCT1 steps are cheaper. Figure 21B discloses rela^ve Tro;er error in the real part of the Loschmidt amplitude, as a func^on of ^me, using one single Tro;er step (le1 panel) and two Tro;er steps(right panel), for different system sizes - = 8,12, ... ,32 (from bo;om to top).Cost comparison with Tro8er.— We now perform a cost comparison against Tro;eriza^on. We saw that as a func^on of ^me, the complexity of our randomized algorithm scales as ^(^^), whereas a Tro;er decomposi^on atfixed Tro;er step # scales as ^(^). In Fig.21A, we show, for sparsity parameter A = 2.3, the different regimes where TETRIS has a lowergate count than one or two Tro;er steps, as a func^on of ^me. We see that, at early ^mes, TETRIS is always cheaper. Moreover, when system size increases, it remains consistently cheaper in theregime where the Loschmidt amplitude is ^ 0.5. At later ^mes, doing one single Tro;er stepalways ends up gebng cheaper. However, increasing ^me at afixed number of Tro;er steps inevitably increases the bias due to the Tro;er error, which we have ignored so far. In Fig.21B, we show the rela^ve Tro;er error as a func^on of ^me, when doing one or two Tro;er steps. We see that in both cases, Tro;eriza^on becomes cheaper only when the Tro;erdecomposi^on incurs at least ≈ 10% error. In par^cular, for ^me < 1, the regime where oneTro;er step is cheaper than TETRIS at op^mal angle would have incurred a Tro;er error of the same order as the error bars on the extrapolated values in Fig.18. Tro;er error would thus have been sta^s^cally visible in Fig.18, even neglec^ng hardware noise. Noise es^ma^on through a mirror-on-average circuit benchmark.— Es^ma^ng the amount of noise in a circuit run on hardware is an important element of quantum compu^ng calcula^ons. However, comparing the measured expecta^on value to exact values is o1en limited to small system sizes or shallow circuits. Scalable methods to es^mate the amount of noise in a circuit are to run a mirror circuit on top, where all the gates are exactly inverted. Since the circuit is logically equivalent to iden^ty, hardware imperfec^ons can be es^mated in a scalable way by es^ma^ng the survival probability of the ini^al state [A35-A37]. These mirror circuit benchmarks, however, are known to overes^mate the noise in actual local observables [A59-A61].Figure 22 discloses numerically simulated mirror-on-average benchmark ^^^^^,^»[Å^$È(h ⊗ z)](E31) (purple), standard mirror benchmark ^^[〈0|MNM|0〉] (teal), both of which return 1 inthe noiseless case, and expecta^on values of the operator ^ = ?b ∑D aD (orange), as a func^onof TQ depolarizing error rate ý7Èc, in size ^ = 12, for %^ = 0.8, taking disorder average of thesparse SYK Hamiltonian ^ . The teal dashed curve indicates the process gatefidelity, 1 −15ý7Èc / 16, to the average number of TQ gates in the circuits, here 1988. The TETRIS algorithm allows instead for a scalable benchmark protocol where a mirror circuit equivalent to iden^ty is implemented only on average. We ini^alize a density matrix Å on theancilla and the system qubits in |+〉 ⊗ |0〉 . We generate two independent random unitariesM, M′ with the TETRIS algorithm, that sa^sfy ^ ^^^^[M] = ^^ . We apply M on the system qubitscondi^oned on the ancilla to be in |0〉, and M′ on the system qubits condi^oned on the ancilla to be in |1〉. The density matrix obtained a1er averaging over M, M′ is 63 MJJ / 168922PCT1 Å^$È: = ^^,^»[Å] =? ^|0〉〈0| ⊗ ^ N áV ^[M|0〉〈0|M ] + ^ |1〉〈0| ⊗ ^^^^|0〉〈0|^^^^^(E31) +áV |0〉〈1| ⊗ ^^^^|0〉〈0|^^^^^ + ?|1〉〈1| ⊗ ^ [N^^ ^» M′|0〉〈0|M′] .Similar as before, deno^ng h ⊗ ^ the observable applying h on the ancilla and ^ on thesystem qubits, we have Tr[Å (h ⊗ ^ ^ ^^^^ ^^^^$È )] = ^ 〈0|^ ^^ |0〉 . (E32)It follows that the ideal expecta^on value of h on the ancilla Tr[Å (h ^^$È ⊗ z)] = ^ is knownexactly. The expecta^on value of h on the ancilla cannot be computed in a scalable way for individual circuits because M and M′ are generally different. However, on averaging overM, M′, a mirror-on-average circuit is implemented, and the expecta^on value of h on the ancillais exactly ^^. Moreover, from the same circuits, the expecta^on value of ^ within ^^^^|0〉 can be computed. We thus expect that the noise observed on the hardware on the expecta^on value of h on the ancilla gives a good es^mate of the noise observed on a local observable. This is in contrast with usual mirror-circuits, which measure a global observable.In Fig.22, we test these ideas with noisy simula^ons of these mirror-on-average circuits. For ^ =12 qubits, and ^me %^ = 0.8 , we plot the expecta^on value of ^ = ?b ∑D aD computed thisway, as well as the expecta^on value of h on the ancilla divided by ^^. We also compare with the expecta^on value obtained when measuring an exact mirror circuit where the inverse MNis applied on the ancilla condi^oned to be 0. All the Hamiltonians are systema^cally averaged over different SYK realiza^ons. Wefirst observe that the standard mirror circuits display a decay with noise that is similar to the circuitfidelity, es^mated as gatefidelity to the number of gates (the agreement is not exact because the number of gatesfluctuates among different random circuits). This is in line with previous implementa^ons of mirror circuit benchmarks [A35-A37]. Secondly, ? we observe that the expecta^on value of a local observable likeb∑DaDis significantly less damped by noise than this mirror circuit. Instead, its decay with noise is similar (although not iden^cal) to that of the mirror-on-average circuits. This agreement between the two is not systema^c, and for other parameter regimes the slope of the mirror-on-average circuits can present more devia^on from than that of observables, but is always smaller than that of the mirror circuits. This shows that these mirror-on-average circuits give a be;er es^mate of noise on actual observables than a standard mirror circuit benchmark, which tends to systema^cally overes^mate the noise on local observables [A35]. Moreover, this mirror-on-average benchmark is scalable in the number of qubits and in the circuit depth. Discussion and outlook.— We have demonstrated an experimental quantum simula^on of the Loschmidt amplitude for the sparse SYK model, based on the TETRIS algorithm. The TETRIS algorithm is par^cularly suited to simulate the SYK model because its disorder average can be readily combined with the random sampling of TETRIS circuits to reduce the sampling overhead. 64 MJJ / 168922PCT1 While the sparsifica^on of the SYK model and the use of TETRIS significantly reduce the circuit complexity, that alo running those circuit proposed two noise the performance of improvements in the accuracy of the experimentally obtained Loschmidt amplitude. In order to address more physically mo^vated ques^ons in future experiments, we discuss the computa^on of an OTOC Tr[8^^^^9^^^^^8N^^^^9N^^^^^], where 8 and 9 are typically set to be Majorana operators. We calcula^on of OTOCs atfinite temperature. Prepara^on of the thermal states at scale requires separate considera^on and is le1 for future work. Here, we replace the trace by the average of the expecta^on values with respect to randomly sampled pure states. With the TETRIS algorithm and the ternary tree encoding [A41], which maps a fermion operator to a spin operator of weight⌈log 2^⌉ , each ^me-evolu^on operator ^±^^^ uses approxim ^ ^K ately ^ ^ logK(2^) ≈2A(%^)^^^logK(2^) TQ gates. To extract the quantum Lyapunov exponent from the f theOTOC, we set the simula^on ^me to %^ ∼ ln-. Therefore, the total TQ gate count is roughly,8 × A(ln(2^)^)^log (2^) ≈ 4 × 10% (^ = 50)K <2 × 10) (^ = 100) (E33)for A = 2.3. Without any paralleliza^on of gates and assuming the exec ^on ^me per circuitdepth being 30 milliseconds (ms), the run^me of each circuit is roughly 4 ×10% × 30 ms = 30hours for ^ = 50. Since ^ / logK(^) ≈ 14 gates can be parallelized by adding the same numberof ancillary qubits, the execu^on ^me per circuit is reduced to two hours. 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Rubin, “Quantum simulation with sum-of-squares spectral amplification,” (2025), arXiv:2505.01528 [quant-ph]. [A63] B. Swingle and M. Winer, Phys. Rev. B 109, 094206 (2024), arXiv:2311.01516 [hep-th]. [A64] M. Hanada, A. Jevicki, X. Liu, E. Rinaldi, and M. Tezuka, JHEP 05, 280 (2024), arXiv:2309.15349 [hep- 11th]. [A65] M. Hanada, S. van Leuven, O. Oktay, and M. Tezuka, “Two-local modifications of SYK model with quantum chaos,” (2025), arXiv:2505.09900 [quant-ph 72 MJJ / 168922PCT1 CLAIMS 1. A computer-implemented method for determining a physical property of a quantum state of a physical system, comprising: genera^ng a computa^onal model represen^ng the physical system, wherein the computa^onal model comprises an ini^al computa^onal state and a ^me-evolu^on operator for performing an evolu^on over ^me to prepare a computa^onal state; genera^ng quantum circuits for implemen^ng the computa^onal model, wherein each of the quantum circuits comprises a sequence of steps to implement the ^me-evolu^on operator, wherein each step of the sequence of steps is drawn randomly from a set of poten^al steps of the ^me-evolu^on operator; execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to output expecta^on values of the ^me evolu^on operator; compu^ng an average of the expecta^on values over the generated quantum circuits; and determining a physical property of the physical system from the computed average of the expecta^on values. 2. A computer-implemented method according to claim 1, wherein the prepared computa^onal state represents an electronic structure state of the physical system, and the determina^on of the physical property comprises a determina^on of the energy of a quantum state of the physical system. 3. A computer-implemented method according to claim 1 or claim 2, wherein the ^me- evolu^on operator performs adiaba^c ^me evolu^on of the prepared computa^onal state. 4. A computer-implemented method according to claim 1, wherein the prepared computa^onal state represents the ground state of the physical system and wherein the operator performs adiaba^c ^me evolu^on of the ground state, and the determina^on of a physical property comprises a determina^on of the ground state energy following adiaba^c ^me evolu^on of the prepared computa^onal state of the physical system. 5. A computer-implemented method according to claim 1, wherein each of the quantum circuits comprises a sequence of rota^ons with a selected gate angle, wherein the rota^ons are each drawn randomly and independent of each other with a rate given by a func^on of the gate angle. 6. A computer-implemented method according to claim 5, wherein the selected gate angle is selected to minimise the expected run^me of the computer-implemented method which run^me is required to reach a selected precision in the determina^on of the physical property. 7. A computer-implemented method according to claims 6, wherein the selected gate angle is selected to minimise the number of shots ^mes the number of gates per shot that is required to reach a selected precision, thereby to minimise the expected run^me. 73

Claims

1. MJJ / 168922PCT1 8. A computer-implemented method according to claim 1, wherein the operator is selected to have a parameter-dependent expecta^on value that is zero when the parameter is equal to the ground state energy of the state, and the method comprises: a1er execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system, measuring the qubits or qudits of the quantum compu^ng system so as to determine the value of the parameter where the parameter-dependent expecta^on value is zero; and outpubng a determina^on of the ground state energy of the quantum state of the physical system.

9. A computer-implemented method according to claim 1, wherein the computa^onal model is a Hamiltonian which represents the states and interac^ons of a physical system.

10. A computer-implemented method according to claim 9, wherein the Hamiltonian is generated as a mathema^cal representa^on of experimentally-determined proper^es of the physical system to be modelled.

11. A computer-implemented method according to claim 9, wherein the Hamiltonian comprises a predefined gate angle parameter, and wherein genera^ng each of the quantum circuits comprises: genera^ng N gates using the Hamiltonian and the predefined gate angle parameter to construct a quantum circuit, genera^ng afirst random number for each of the N gates wherein thefirst random number determines the number of ^mes each of the N gates is applied in the quantum circuit, and genera^ng a second random number for each gate to be applied in the quantum circuit, wherein the second random number determines the order that each gate is applied in the quantum circuit.

12. A computer-implemented method according to claim 11, to be performed on a computer system comprising a quantum computer having a plurality of qubits or qudits and a controller for applying the N gates using a combina^on of the qubits or qudits, wherein the controller is adapted to apply at least some of the N gates between non-nearest neighbour qubits or qudits of the plurality of qubits or qudits.

13. A computer-implemented method according to claim 11, wherein thefirst random number is generated from a Poisson distribu^on of numbers.

14. A quantum compu^ng system comprising a quantum computer having a plurality of qubits or qudits for execu^ng quantum circuits, and further comprising a controller for controlling performance of opera^ons on the quantum computer, to perform a method comprising: genera^ng a computa^onal model represen^ng the physical system, wherein the computa^onal model comprises an ini^al computa^onal state and a ^me-evolu^on operator for performing an evolu^on over ^me to prepare a ^me-evolved computa^onal state; genera^ng quantum circuits for implemen^ng the computa^onal model, which quantum circuits each comprise a sequence of gate opera^ons having a selected gate angle, to implement the ^me-evolu^on operator, wherein each gate opera^on of the sequence of gate opera^ons is 74 MJJ / 168922PCT1 drawn randomly from a set of opera^ons of the ^me-evolu^on operator; execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to output expecta^on values of the ^me evolu^on operator; compu^ng an average of the expecta^on values over the generated quantum circuits; and rescaling the computed average by a factor that is a func^on of the gate angle; and determining a physical property of the physical system based on the rescaled computed average.

15. A quantum compu^ng system according to claim 14, configured to allow gate opera^ons between non-adjacent qubits of the plurality of qubits or qudits.

16. A quantum compu^ng system according to claim 15, configured for any-to-any connec^vity between the plurality of qubits or qudits.

17. A computer-implemented method for determining the energy of a quantum state of a physical system, comprising: genera^ng a computa^onal model represen^ng the energy of a quantum state of the physical system, wherein the computa^onal model comprises an ini^al computa^onal state and a ^me-evolu^on operator for performing an evolu^on over ^me to prepare a ^me-evolved computa^onal state, wherein the operator has a parameter that encodes the energy of the ^me- evolved computa^onal state and the operator has a parameter-dependent expecta^on value that is zero when the parameter is equal to the ground state energy of the quantum state; genera^ng quantum circuits corresponding to the computa^onal model; execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system to compute the effects of the ^me-evolu^on operator of the computa^onal model; and measuring the qubits or qudits of the quantum compu^ng system so as to determine the value of the parameter where the parameter-dependent expecta^on value is zero, and using the determined parameter value to determine the ground state energy of the quantum state of the physical system.

18. A computer-implemented method according to claim 17, wherein the step of genera^ng quantum circuits comprises genera^ng quantum circuits that each comprises a sequence of opera^ons of the ^me-evolu^on operator, wherein each opera^on of the sequence of opera^ons is drawn randomly from a set of opera^ons of the ^me-evolu^on operator.

19. A computer-implemented method according to claim 17, wherein the step of genera^ng quantum circuits comprises genera^ng quantum circuits that each comprise rota^ons for each of a plurality of terms in the computa^onal model, the rota^ons for each term comprising a sequence of rota^ons of a selected gate angle, wherein the rota^ons are each drawn randomly and independently of each other with a rate given by a func^on of the gate angle.

20. A computer-implemented method according to claim 17, wherein the ^me-evolu^on operator performs adiaba^c evolu^on of the prepared computa^onal state. 75 MJJ / 168922PCT1 21. A computer-implemented method according to claim 20, wherein the prepared computa^onal state represents the ground state of the physical system and wherein the operator performs adiaba^c ^me evolu^on of the ground state, and the output determina^on is a determina^on of the ground state energy following adiaba^c ^me-evolu^on of the prepared computa^onal state of the physical system.

22. A computer-implemented method according to claim 17, wherein the operator is selected to have a parameter-dependent expecta^on value that is zero when the parameter is equal to the ground state energy of the state, and the method comprises: a1er execu^ng the quantum circuits on the qubits or qudits of a quantum compu^ng system, measuring the qubits or qudits of the quantum compu^ng system so as to determine the value of the parameter where the parameter-dependent expecta^on value is zero; and outpubng a determina^on of the ground state energy of the quantum state of the physical system.

23. A computer-implemented method according to claim 17, wherein the computa^onal model is a Hamiltonian which represents the states and interac^ons of a physical system.

24. A computer-implemented method according to claim 23, wherein the Hamiltonian is generated as a mathema^cal representa^on of experimentally-determined proper^es of the physical system to be modelled.

25. A computer-implemented method according to claim 23, wherein the Hamiltonian comprises a predefined gate angle parameter, and wherein genera^ng each of the quantum circuits comprises: genera^ng N gates using the Hamiltonian and the predefined gate angle parameter to construct a quantum circuit, genera^ng afirst random number for each of the N gates wherein thefirst random number determines the number of ^mes each of the N gates is applied in the quantum circuit, and genera^ng a second random number for each gate to be applied in the quantum circuit, wherein the second random number determines the order that each gate is applied in the quantum circuit.

26. A computer-implemented method according to claim 25, to be performed on a computer system comprising a quantum computer having a plurality of qubits or qudits and a controller for applying the N gates using a combina^on of the qubits or qudits, wherein the controller is adapted to apply at least some of the N gates between non-nearest neighbour qubits or qudits of the plurality of qubits or qudits. 76 MJJ / 168922PCT1 ABSTRACT Provided are computer-implemented methods and quantum computing systems for preparing computational states representing quantum states of a physical system, including performing a computational evolution of the state and then determining physical properties of the system using the time-evolved computational state. A described computer-implemented method enables computing Hamiltonian dynamics of observables on a quantum computer to provide information about the physical system represented by the Hamiltonian. A described computer-implemented method prepares low energy electronic structure states of a physical system using adiabatic evolution. The method determines the energy of an equilibrium quantum state of a physical system using a time evolution operator that enables adiabatic evolution of a prepared electronic structure state on a quantum computing system. A described method involves indirectly determining the energy of an eigenstate of a physical system by evaluating the expectation values of a time- evolution operator averaged across multiple shots for randomly-generated quantum circuits. Example methods enable calculation of the energy of an evolved computational state with chemical accuracy, due to avoiding discretization errors, with a smaller circuit depth than known alternatives. 77

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