Efficient fault tolerant simulation of

Abstract

The invention discloses an efficient fault-tolerant Trout simulation of molecular Hamiltonian. A method performed by a quantum computing device, the method comprising: applying a quantum circuit to a qubit system included in the quantum computing device, where the quantum circuit implements a time-evolved Trout step, comprising: applying a multi-layer quantum logic gate to the qubit system, where the quantum logic gate comprises a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step of a Trout step; each application of a layer in the multilayer evolves the qubit system under unitary operators generated by respective sets of interaction terms in a qubit Hamiltonian representing the Fermi system, and each layer in the multilayer quantum logic gate includes rotation operations having the same rotation angle, applying a layer of a quantum logic gate to the qubit system includes correspondingly applying a Hamming weight to a rotation in the layer of the quantum logic gate; and measuring the evolved qubit system to determine the energy of the Fermi system.

Description

[0001] This application is a divisional application of the invention patent application with the application date of August 13, 2019, Chinese application number 201980063375.3, and invention name “Efficient Fault-Tolerant Trotter Simulation of Molecular Hamiltonian”. Technical Field

[0002] This specification relates to quantum computing. Background Art

[0003] A quantum simulator is a device designed to provide information and insights about a physical system or device. Quantum simulators enable the simulation of physical systems that are difficult to study in the laboratory or impossible to model using classical processors. Summary of the Invention

[0004] This specification describes techniques for performing error-tolerant Trotter simulations of molecular Hamiltonians.

[0005] In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for determining a property of a physical system, the method comprising: transforming a Hamiltonian describing the physical system into a qubit Hamiltonian describing a corresponding quantum bit (qubit) system, wherein the qubit Hamiltonian comprises a plurality of two-qubit interaction terms, each two-qubit interaction term comprising a corresponding translation-invariant two-qubit interaction term coefficient; evolving the qubit system under a unitary operator produced by the plurality of two-qubit interaction terms, comprising: applying multiple layers of quantum logic gates to the qubit system, wherein each application of a layer in the multiple layers evolves the qubit system under a unitary operator produced by a corresponding subset of the plurality of two-qubit interaction terms; wherein the values ​​of the two-qubit interaction term coefficients for the subset of the plurality of two-qubit interaction terms that produce the unitary operator are constant; measuring the evolved qubit system; and determining one or more properties of the physical system based on the measurement of the evolved qubit system.

[0006] Other embodiments of this aspect include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each of which is configured to perform the actions of the method. One or more systems of classical and / or quantum computers can be configured to perform specific operations or actions by means of software, firmware, hardware, or a combination thereof installed on the system, which software, firmware, hardware, or a combination thereof causes the system to perform the actions when in operation. One or more computer programs can be configured to perform specific operations or actions by means of including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0007] The foregoing and other embodiments may optionally include one or more of the following features, alone or in combination: In some embodiments, each layer in a multi-layer quantum logic gate includes a rotation operation having the same rotation angle.

[0008] In some embodiments, the method further includes generating a multi-layer quantum logic gate, the generating the multi-layer quantum logic gate comprising, for a system with an even number N qubits: arranging the N qubits into a two-dimensional grid of qubits, wherein the position of each qubit in the grid is represented by a position vector p; for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p over all N qubits; and generating two layers of qubit interactions based on the defined set of qubit pairs, wherein each qubit in each layer interacts with another qubit.

[0009] In some embodiments, each layer in a multi-layer quantum logic gate includes N / 2 isometric rotation operations.

[0010] In some embodiments, the method further includes generating a multi-layer quantum logic gate, the generating the multi-layer quantum logic gate comprising, for a system with an odd number N qubits: arranging the N qubits into a two-dimensional grid of qubits, wherein the position of each qubit in the grid is represented by a position vector p; for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p across all N qubits, wherein one qubit is not included in the defined set of qubit pairs; and generating three layers of qubit interactions based on the defined set of qubit pairs, comprising: a first layer in which each qubit interacts with another qubit, a second layer in which each qubit interacts with another qubit, and a third layer comprising a single rotation operation.

[0011] In some embodiments, the first and second layers of the quantum logic gate each include (N-1) / 2 isometric rotation operations.

[0012] In some embodiments, evolving the qubit system under a unitary operator generated by the plurality of two-qubit interaction terms further comprises, in each layer of the plurality of layers, assigning a Hamming weight to a rotation in the layer.

[0013] In some embodiments, defining a set of qubit pairs (p, p+s) for each qubit p includes evaluating p+s modulo a grid length.

[0014] In some embodiments, the qubit Hamiltonian further includes a kinetic energy term and a single-qubit potential energy term, and wherein the method further includes evolving the qubit system under a unitary operator generated by the kinetic energy term and a unitary operator generated by the single-qubit potential energy term.

[0015] In some embodiments, evolving a qubit system under a unitary operator resulting from a plurality of two-qubit interaction terms, a kinetic energy term, or a single-qubit potential energy term comprises implementing time evolution or implementing directionally controlled evolution on an auxiliary bit.

[0016] In some embodiments, evolving the qubit system under a unitary operator resulting from a plurality of two-qubit interaction terms, a kinetic energy term, or a single-qubit potential energy term comprises evolving the qubit system in a split-operator Trotter step.

[0017] In some embodiments, the splitting operator Trotter step is a step among a plurality of splitting operator Trotter steps that produces a minimum Trotter error.

[0018] In some embodiments, the Hamiltonian describing the physical system comprises an electronic structure Hamiltonian.

[0019] In some embodiments, the physical system includes a chemical substance or material, and wherein determining a property of the physical system based on measurements of the evolving qubit system includes determining the property of the chemical substance or material using a simulated evolution of the qubit system under a qubit Hamiltonian.

[0020] Embodiments of the present disclosure provide a method performed by a quantum computing device, the method comprising: applying a quantum circuit to a qubit system included in the quantum computing device, wherein the quantum circuit implements a time-evolved Trotter step, comprising: applying a multi-layer quantum logic gate to the qubit system, wherein each application of a layer in the multi-layer evolves the qubit system under a unitary operator generated by a corresponding set of interaction terms in a qubit Hamiltonian representing a Fermi system, and each layer in the multi-layer quantum logic gate comprises a rotation operation having the same rotation angle, wherein applying the layers of the quantum logic gate to the qubit system comprises assigning Hamming weights corresponding to the rotations in the layers of the quantum logic gate; and measuring the evolved qubit system to determine the energy of the Fermi system.

[0021] An embodiment of the present disclosure provides an apparatus comprising: quantum computing hardware comprising: a qubit system; and a control device configured to apply a quantum circuit of quantum logic gates to qubits included in the qubit system and measure the qubits included in the qubit system; wherein the apparatus is configured to perform operations comprising: applying a quantum circuit to the qubit system included in the quantum computing device, wherein the quantum circuit implements a time-evolved Trotter step comprising: applying a multilayer quantum logic gate to the qubit system, wherein each application of a layer in the multilayer evolves the qubit system under a unitary operator generated by a corresponding set of interaction terms in a qubit Hamiltonian representing a Fermi system, and wherein each layer in the multilayer quantum logic gate comprises a rotation operation having the same rotation angle, wherein applying the layers of quantum logic gates to the qubit system comprises assigning Hamming weights to the rotations in the layers of the quantum logic gates accordingly; and measuring the evolved qubit system to determine the energy of the Fermi system.

[0022] An embodiment of the present disclosure provides a method for determining a property of a physical system, the method being performed by a system comprising one or more classical processors and quantum hardware, the quantum hardware comprising: a system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits, the method comprising: generating a multilayer quantum logic gate, wherein each layer in the multilayer quantum logic gate comprises a rotation operation having the same rotation angle, the generating comprising: for a system of N even-numbered qubits and for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p on all N qubits, wherein the position of each qubit is represented by the position vector p, and a set of qubit pairs based on the definition The qubit pairs generate two layers of qubit interactions, wherein each qubit in each layer interacts with another qubit; or for a system of N odd number of qubits and for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p over all N qubits, wherein one qubit is not included in the defined set of pairs, and generating three layers of qubit interactions based on the defined set of qubit pairs, the three layers comprising: i) a first layer, wherein each qubit interacts with another qubit, ii) a second layer, wherein each qubit interacts with another qubit, and iii) a third layer comprising a single rotation operation; a system for applying the multi-layer quantum logic gate to qubits; and a system for measuring qubits by the quantum hardware to determine one or more properties of the physical system.

[0023] An embodiment of the present disclosure provides an apparatus for determining a property of a physical system, the apparatus comprising: one or more classical processors; quantum hardware comprising: a system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits; wherein the apparatus is configured to perform a method according to an embodiment of the present disclosure.

[0024] Embodiments of the present disclosure provide a method for determining a property of a physical system, the method being performed by a system including one or more classical processors and quantum hardware, the quantum hardware including: a system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits, the method comprising: generating a multi-layer quantum logic gate, wherein each layer in the multi-layer quantum logic gate evolves the system of interacting qubits under a corresponding unitary operator generated by a corresponding set of two-qubit interaction terms included in a qubit Hamiltonian describing the physical system and includes rotation operations having the same rotation angle; combining the rotation operations within each layer in the multi-layer quantum logic gate using Hamming weight phasing to generate a plurality of update layers of the quantum logic gate; applying the plurality of update layers of the quantum logic gate to the system of qubits; and measuring the qubits by the quantum hardware to determine one or more properties of the physical system.

[0025] Embodiments of the present disclosure provide an apparatus for determining a property of a physical system, the apparatus comprising: one or more classical processors; quantum hardware comprising: a system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits; wherein the apparatus is configured to perform a method according to an embodiment of the present disclosure.

[0026] The subject matter described in this specification can be implemented in a specific way to realize one or more of the following advantages.

[0027] The presently described disclosure represents a significant and broadly applicable improvement over the prior art of quantum simulation.

[0028] When determining (e.g. simulating) the evolution of the electronic structure Hamiltonian using Trotter-based methods, the evolution of the interaction part of the potential operator typically appears to require applying O(N) 2) rotations. The technique described in this application reduces the number of required applications of rotations to O(NlogN) at the cost of 4(N-1)(N-2) additional T-gates. Since the number of T-gates required to synthesize each rotation is much greater than 4, this greatly reduces the error-tolerance cost of determining these terms.

[0029] Furthermore, the techniques described in this specification can be applied to any quantum simulation of the electronic structure Hamiltonian using a plane wave dual basis, or more generally to any simulation of a two-qubit interaction term with the translation-invariant property of equation (3) below. Thus, a variety of different settings can benefit from the described techniques. For example, the techniques can be used to more efficiently design and test materials or catalysts, more accurately predict the rates of certain chemical reactions, or more efficiently synthesize drugs.

[0030] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 Depicts an example system for determining properties of a physical system.

[0032] Figure 2 is a flow chart of an example process for determining properties of a physical system.

[0033] Figure 3A is a flow chart of an example process for determining the interactions in each layer of a multi-layer quantum logic gate applied to an even number of qubits.

[0034] Figure 3B is a flow chart of an example process for determining interactions in each layer of a multi-layer quantum logic gate applied to an odd number of qubits.

[0035] Figure 4 is a diagram of two example layers of interacting qubits.

[0036] Like reference numbers and designations in the various drawings indicate like elements. DETAILED DESCRIPTION

[0037] The electronic structure Hamiltonian describes the properties of electrons interacting in the presence of a fixed nucleus. The physics of these systems determines the rates of chemical reactions and the properties of most materials. Therefore, determining the electronic structure Hamiltonian has applications in a wide range of fields, from drug synthesis to the design of novel catalysts and materials.

[0038] The dynamics of these systems are often intractable for classical computers. Consequently, a variety of methods have been developed for quantum simulations of electronic structures, including second quantization, first quantization, algorithms in real space; adiabatic algorithms; methods based on the Taylor series method of time-evolution and other algorithms using linear combinations of unitary operator frameworks; variational algorithms; Bravyi-Kitaev coding; reduced locality Jordan-Wigner coding and low-rank tensor decomposition of Coulomb operators. However, the only methods that have been compiled all the way down to the elementary gate set and have been rigorously evaluated for feasibility within error correction are Trotter-based methods in a second-quantized basis.

[0039] This specification describes new and improved techniques that build upon such Trotter-based methods. The present techniques reduce the number of rotation operations required to achieve evolution under the interaction part of the potential operator in the split-operator Trotter step. Typically, evolution under the interaction part of the potential operator requires O(N) for N qubits. 2 However, by working in a plane wave dual basis and constructing ordered layers of quantum logic gates in which equiangular rotations are grouped together, it is possible to apply Hamming weight phasing and reduce the evolutionary cost under the interaction part of the potential operator to O(NlogN) rotations.

[0040] Example Hardware

[0041] Figure 1 An example system 100 is depicted for determining a physical system described by an electronic structure Hamiltonian. Example system 100 is an example of a system implemented as a classical or quantum computer program on one or more classical computers or quantum computing devices at one or more locations, and in which the systems, components, and techniques described below may be implemented.

[0042] System 100 may include quantum hardware 102 in data communication with a classical processor 104. System 100 may receive data as input, which may include data representing a physical system of interest, such as input data 106. The received data representing the physical system of interest may include data representing a Hamiltonian that describes the physical system to be modeled or simulated. In some embodiments, the received data may include data representing an electronic structure Hamiltonian that describes a corresponding physical system, such as a single molecule, material, or chemical substance.

[0043] The system may generate as output data representing the results of determining a physical system of interest, for example, output data 108 representing determined physical properties of the physical system. For example, as described above, in some embodiments, the physical system may be a chemical substance. In these cases, the data representing the simulation results may be used to determine properties of the chemical substance, such as the rate of a chemical reaction. As another example, also described above, in some embodiments, the physical system may be a material, such as a semiconductor. In these cases, the data representing the simulation results may be used to determine properties of the material, such as electrical conductivity.

[0044] System 100 is configured to perform classical computations in conjunction with quantum computations using a classical processor 104 and quantum hardware 102. Classical processor 104 may include a Hamiltonian transformation module 114 configured to process received input data representing a Hamiltonian describing a physical system, such as input data 106, and generate data representing a qubit Hamiltonian describing a qubit system corresponding to the physical system. For example, Hamiltonian transformation module 114 may be configured to apply a Jordan-Wigner transform to received input data 106 to generate data representing the corresponding qubit Hamiltonian.

[0045] The quantum hardware 102 is configured to determine the evolution of a physical system specified by the input data 106, for example, by simulation using corresponding data representing a qubit Hamiltonian describing the qubit system. For example, as described below with reference to Figure 2-4 As described in detail, quantum hardware 102 may be configured to implement a Trotter step, such as a split-operator Trotter step 118, by applying a quantum circuit that affects a unitary operator that depends on the qubit Hamiltonian to the state of a system of qubits 110 included in the quantum hardware.

[0046] The type of qubits 110 included in quantum hardware 102 depends on a variety of factors, such as the type of physical system to be determined or simulated or laboratory limitations, and can vary. For example, in some cases, it may be convenient to include one or more resonators attached to one or more superconducting qubits (e.g., Gmon or Xmon qubits). In other cases, ion traps, photonic devices, or superconducting cavities (with which states can be prepared without qubits) can be used. Other example implementations of qubits include fluxmon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0047] The qubits 110 may be arranged in a grid with controllable interactions between the qubits. One or more control devices 112 include devices such as arbitrary waveform generators that are configured to apply quantum circuits of quantum logic gates (e.g., rotation operations) to the qubits 110, for example, by applying corresponding control pulses along control lines that couple the qubits 110 to the control devices 112. The control devices 112 may also include devices such as readout resonators that are configured to perform measurements on the qubits 110 and provide the measurement results 120 to the classical processor 104.

[0048] As the following reference Figure 2 As detailed in step 208 of , the measurements 120 received by the classical processor 104 may be processed to determine properties of the physical system specified by the input data 106 .

[0049] Programming the hardware

[0050] Figure 2 is a flow chart of an example process 200 for determining a property of a physical system. For convenience, process 200 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system, such as Figure 1 The system 100 may perform the process 200 .

[0051] The system transforms the Hamiltonian describing the physical system into a qubit Hamiltonian describing the corresponding qubit system (step 202). For example, the Hamiltonian describing the physical system can be a Fermi Hamiltonian of the following form:

[0052]

[0053] Among them, T pq 、U p and V pqrepresents the number defined by integrals over molecular orbitals, due to kinetic energy, external potential, and electron-electron Coulomb repulsion, and the creation and annihilation operators a p 、 The vectors p and q are used to index, for example, lattice sites and / or spin orbitals. pq 、U p and V pq In some embodiments, T pq and / or V pq One or more of can be plural. In some embodiments, the Hamiltonian H can be a Hamiltonian describing a material (e.g., a polymer airplane wing or rocket, a solar cell, a battery, a catalytic converter, or a thin-film electronic device), or a Hamiltonian describing a chemical substance.

[0054] The Jordan-Wigner transformation can be used to transform the Hamiltonian in the form of equation (1) into a qubit Hamiltonian describing the corresponding qubit system. The Jordan-Wigner transformation transforms the Fermi annihilation and production operators and a p Transformed into the Pauli operator X corresponding to the spin 1 / 2 particle p 、Y p 、Z p The standard form of the Jordan-Wigner transformation is given by and Transforming the Hamiltonian of equation (1) using the Jordan-Wigner transformation yields the qubit Hamiltonian given below in equation (2), where the identity term is omitted due to its ease of determination.

[0055]

[0056] In equation (2), X p 、Y p and Z p denote the Pauli X, Y, and Z gates, respectively, acting on qubits indexed by p in the Jordan-Wigner ordering. represents a chain of Pauli Z-gates between qubit p and qubit q in the Jordan-Wigner order (needed to preserve the appropriate commutation relation of the transformed Fermi creation and annihilation operators) and and denotes coefficients that can be directly computed from the corresponding coefficients in the second quantized Hamiltonian given by equation (1) above.

[0057] The Hamiltonian of equation (1) can be expressed in one of a number of different bases. For convenience, in equation (1), the Hamiltonian H is written in a plane wave basis. However, one or more terms of the Hamiltonian H can also be expressed in a plane wave dual basis. A plane wave dual basis can be obtained by applying a unitary discrete Fourier transform to a plane wave basis.

[0058] In the plane wave dual basis, the qubit Hamiltonian H qubit Including multiple translation-invariant coefficients The two-qubit interaction term That is, for any vector s that indexes a particular qubit in the qubit system:

[0059]

[0060] To determine properties of the physical system, the system uses the qubit system and the qubit Hamiltonian to perform a quantum simulation of the evolution of the physical system under the Hamiltonian H. Performing the quantum simulation involves performing a quantum transformation on the unitary operator generated by the qubit Hamiltonian using a variation of the split-operator algorithm. The evolution of the qubit system under Hamiltonian H is determined.

[0061] The splitting operator algorithm can be used to transform the Hamiltonian H qubit Evolution under Split into separate evolutions in the kinetic and potential parts of the Hamiltonian. This is particularly useful because the terms in the kinetic part of the Hamiltonian (with coefficients ) can be diagonalized via an efficient circuit transformation C, leading to a sum of single-qubit terms, namely:

[0062]

[0063] Diagonalization circuit C, Use the fast Fermi Fourier transform or Givens rotation to change between position and momentum basis. In the splitting operator algorithm, the Trotter-Suzuki approximation can be applied to convert the total Hamiltonian H qubit The evolution under the condition of The first evolution under e -iTt and potential energy terms The second evolution under e -i(U+V)t .

[0064] Symmetrize these evolutions to give a second-order Trott step by the circuit C and its inverse Separate evolution blocks to approximate the time evolution of the total Hamiltonian

[0065]

[0066] Equation (4) describes a single second-order Trotter step. The approximation can be refined by dividing t into such steps, each with time t / r.

[0067] There are several options for how to symmetrize the splitting operator Trotter step. Instead of using equation (4), which approximates the evolution of r Trotter steps as e -iHt ≈(e -iTt / 2r e -i(U+V)t / r e -iTt / 2r ) r , an alternative choice is to approximate the evolution of r Trotter steps as e -iHt ≈(e -i(U+V)t / 2r e -iTt / r e -i(U+V)t / 2r ) r , where the single-order Trott step of this form is given by equation (5) below.

[0068]

[0069] When different circuits are implemented in the splitting operator steps given by equations (4) or (5), each term or An arbitrary rotation is required, either to achieve time evolution or in the form of auxiliary bits that can be used for phase estimation. Thus, with N spin orbits, at most N(N+3) / 2 rotations are required to determine the evolution under all terms in the Hamiltonian, and this can be achieved by a single application of the basis change circuit C and its inverse Each of the steps in the circuit completes the Trotter.

[0070] At first glance, the number of gates required for the two Trotter steps given by equations (4) and (5) may appear to be different. The number of terms in T is linear in the number of spin-orbits, while the number of terms in U+V is quadratic in the number of spin-orbits. Therefore, it can be expected that with e -i(U+V)t Equation (5) appears twice more than the equation (5) in which e -i(U+V)t Equation (4) that appears only once requires more gates. On the other hand, in Equation (4) C or appears four times, but only twice in equation (5), which may indicate that equation (4) can use more gates.

[0071] However, this intuition is deceptive. Since the number of Trotter steps r is typically very large, e.g., r > 400, and the beginning and end of the two splitting operator steps are merged together, the cost difference is negligible. As a result of this merging, the cost of the r-Trott step simulation is almost the same for the two competing orderings of the terms. In the final Trotter step, the two orderings differ: Equation (4) applies C once more than in Equation (5), whereas e is applied once more. -i(U+V)t Both variants apply C in each of the remaining r-1 Trotter steps. e -iTt t and e -i(U+V)t For once, that final difference is completely outweighed.

[0072] Instead, the difference in the number of Trotter steps r required to perform an accurate simulation (determined by the Trotter errors of the two steps) determines the cost difference. Thus, when executing the example process 200, the Trotter step configuration selected depends on which splitting operator step produces the smaller Trotter error.

[0073] When the unitary operator given by equations (4) and (5) above When the evolution of the qubit system is realized under the condition of The evolution of the resulting unitary operator allows for the determination (e.g., simulation) of The value of each of the two-qubit interaction terms is the same (step 204). That is, the system applies a multi-layer quantum logic gate to the qubit system, wherein each application of a layer in the multi-layer causes the qubit system to evolve under a unitary operation produced by a corresponding subset of the plurality of two-qubit interaction terms, and the values ​​of the two-qubit interaction term coefficients of the subset of the plurality of two-qubit interaction terms that produce the unitary operator are constant. Below, with reference to FIG. 3 and Figure 4 Methods for selecting the interactions simulated in each layer of a multilayer quantum logic gate are described in more detail.

[0074] By this way The rotation operations used in each layer all have the same rotation angle. This enables the rotation operations within each layer to be fully combined using Hamming weight phasing. Hamming weight phasing reduces M parallel rotations of the same angle to M parallel rotations of different angles using M-1 auxiliary qubits. rotations, and 4M-4 additional T-gates. Since synthesizing arbitrary rotations on a fault-tolerant quantum computer is very expensive, this can result in significant cost savings. An example application of Hamming weight phasing is described below.

[0075] The system measures the evolving qubit system (step 206). In some embodiments, this can include measuring the phase accumulated during the evolution determined on the individual auxiliary qubits. For example, this can allow determination of the electronic ground state energy for a single value of a set of nuclear coordinates.

[0076] Based on measurements of the evolving qubit system, the system determines one or more properties of the physical system (step 208).

[0077] For example, in some embodiments, the Hamiltonian describing the physical system can represent a chemical species. In these embodiments, the system can perform steps 204 and 206 for each nuclear coordinate (depending on the specific chemical species represented by the Hamiltonian), where measuring the evolving qubit system includes measuring the energy as a function of each nuclear coordinate. Such measurements can be used to generate a potential energy surface from which the rate of the chemical reaction can be determined.

[0078] As another example, in some embodiments, a Hamiltonian describing a physical system can characterize a material. In these embodiments, a system can measure the electronic ground state energy and use the ground state energy to determine physical properties of the material. For example, information about the equilibrium crystal structure of a material can be determined by performing measurements on multiple evolving qubit systems, where each system represents a different set of nuclear coordinates for the material. From these measurements, a ground state energy can be determined for each set of nuclear coordinates. The lowest ground state energy determined from these measurements can correspond to a crystal structure for the material. Other ground state energies can be used to determine other physical properties of the system, such as phase transitions between different structures. Other properties that can be determined for a material by simulating a Hamiltonian in this manner include: magnetic order; electronic structure; electrical properties of the material, such as conductivity; and / or mechanical properties, such as Young's modulus and / or compressibility.

[0079] Figure 3A is a flow chart of an example process 300 for determining interactions in each layer of a multi-layer quantum logic gate applied to an even number of qubits. For convenience, process 300 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 300 .

[0080] The system arranges N qubits into a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p (step 302). Figure 4A diagram of an example 4×4 grid of 16 qubits is shown in . The position vector is given by a two-dimensional vector. In some embodiments, the components of the vector are integers that mark the positions in the two-dimensional grid of qubits.

[0081] For each index vector s, the system defines a set of qubit pairs {(p, p+s)} for each position vector p across all qubits (step 304). For all qubit position vectors p, the coefficients The translation invariance of guaranties:

[0082]

[0083] When the number of qubits N is even, there are N such qubit pairs (p, p+s), which are given by N-1 different index vectors s on the grid. However, since each of these interactions is between two qubits, at most N / 2 interactions can be performed simultaneously in a layer. Therefore, for each index vector s, the system generates two layers of qubit interactions based on a defined set of qubit pairs, where each qubit in each layer interacts with another qubit (step 306). In each layer, N / 2 interactions of equal strength can be simulated, resulting in N / 2 equiangular rotations that can be combined using Hamming weight phasing. Figure 4 Two example layers of interacting qubits with an index vector s = (0, 1) are shown in FIG. The index vector is given by a two-dimensional vector. In some embodiments, the components of the vector are integers that represent the position difference between two qubits in the two-dimensional grid of qubits, for example, s = (m, n), where m and n are integers representing the position difference between the two qubits along two perpendicular axes.

[0084] In general, there are N-1 different index vectors s that must be iterated, since the final vector s = 0 will not occur because it corresponds to a qubit interacting with itself. Each index vector s is applied in two layers, resulting in 2N-2 layers, each determining N / 2 interactions. In each of the 2N-2 layers, one can Hamming weight phasing is applied on are the same, and therefore each of the N / 2 rotation angles is the same. This allows the Hamming weight regularization to be applied to each rotation in each layer, reducing the number of rotations required per layer from N / 2 to , at the cost of an additional 2N-4 T-gates. The T-cost for all two-qubit terms in the steps used to determine the Trotter evolution over all 2N-2 layers is Among them, T synth=O(log(1 / ∈)) represents the number of T-gates required to synthesize each rotation, where ∈ represents the required synthesis accuracy.

[0085] Figure 3B is a flow chart of an example process 350 for determining interactions in each layer of a multi-layer quantum logic gate applied to an odd number of qubits. For convenience, process 350 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform process 350 .

[0086] The system arranges the N qubits into a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p (step 352).

[0087] For each index vector s, the system defines a set of qubit pairs {(p, p+s)} for each position vector p across all qubits (step 354). Because the number of qubits is odd, one qubit will not be included in the defined set of qubit pairs. Similarly, for all qubit position vectors p, the coefficients The translation invariance of

[0088] When the number of qubits N is odd, three layers are used instead of two to simulate each of the N-1 index vectors s. This is because, when the number of qubits is odd, one qubit is left over in each layer when defining qubit pairs. As a result, there are 2N-2 layers of (N-1) / 2 equiangular rotations and N-1 layers of individual rotations. Hamming weight phasing can then be applied to reduce the (N-1) / 2 equiangular rotations per layer to s at the cost of an additional 2N-2 T-gates. The T-cost for all two-qubit terms in the steps used to determine the Trotter evolution over all levels is

[0089] For both even and odd N, use at most 4(N-1) 2 The extra T-gates reduce the cost of the rotation to O(NlogNlog(1 / ∈)). This is comparable to the O(NlogNlog(1 / ∈)) cost when no Hamming weight phasing is applied or the interaction term is not translation invariant. 2 The improvement applies to any quantum simulation of electronic structure using a plane-wave dual basis, or more generally to any simulation of two-qubit interactions with the translation-invariant property given by Equation (3).

[0090] Figure 4Figure 400 shows two example layers (a) and (b) of qubit interactions. For the case of 16 qubits on a 4×4 grid, two example layers are generated for index vector s = (0, 1). In both layers, qubit p interacts with qubit p + s, where the elements of index vectors p and p + s are evaluated modulo the grid length. For example, considering the first row 410 of the 4×4 grid in layer (a), qubit 402 at position p = (0, 0) interacts with qubit 404 at position p + s = (0, 0) + (0, 1) = (0, 1), and qubit 406 at position p = (0, 2) interacts with qubit 408 at position p + s = (0, 2) + (0, 1) = (0, 3). In layer (b), qubit 404 at position p = (0, 1) interacts with qubit 406 at position p + s = (0, 1) + (0, 1) = (0, 2), and qubit 408 at position p = (0, 3) interacts with qubit 402 at position p + s = (0, 3) + (0, 1) = (0, 4) ≡ (0, 0) mod (4).

[0091] Since each interaction (indicated by a dashed line, e.g., line 412) is between two qubits, a maximum of [N=16] / 2=8 interactions can be performed simultaneously in a layer. Thus, for index vector s=(0,1), two layers (a) and (b) of 8 qubit interactions are generated, where each qubit in each layer interacts with another qubit.

[0092] Example Applications of Hamming Weight Phasing

[0093] Consider rotating the same single qubit by R z (θ) = exp(-iθZ / 2) is applied simultaneously to a circuit with three different qubits. The actions on the logical states are of different phases, depending only on the Hamming weight of the logical state. Specifically,

[0094] 1. All zero state |000>take phase -3θ / 2,

[0095] 2. The three states |001>, |010>, |100> of Hamming weight 1 each take a phase of -θ / 2.

[0096] 3. The three states |011>, |101>, |110> of Hamming weight 2 are all phased by θ / 2, and

[0097] 4. The all-1 state |111> takes a phase of 3θ / 2.

[0098] The Hamming weight of the input state can be calculated without applying the same angle R z(θ), and two different rotations can be applied to the Hamming weights: z (θ) is applied to 1 bit (1s bit), and R z (2θ) is applied to 2 bits (2sbit). In this case,

[0099] 1. Applying a phase of -3θ / 2 to the all-zero state |000> (Hamming weight 0),

[0100] 2. Apply -θ / 2 = -θ + θ / 2 to |001>, |010>, |100> (in binary, the Hamming weight is 1 = 01 b ),

[0101] 3. Apply θ / 2 = +θ - θ / 2 to |011>, |101>, |110> (Hamming weight 2 = 10 b ),as well as

[0102] 4. Applied to all 1 states |111> (Hamming weight 3 = 11 b ).

[0103] For both processes, the phase at each logic state is the same. However, since the rotations must be synthesized using expensive T-gates, reducing the number of rotation gates in the circuit reduces its fault tolerance cost.

[0104] This idea is easily extended to n repeated equiangular rotations R occurring in parallel in a circuit. z (θ) case: instead of applying n original rotations, one can compute the Hamming weights of the relevant qubits and instead apply Rotation R z (θ), R z (2θ), R z (4θ), … are applied to the Hamming weights. This technique is called Hamming weight phasing.

[0105] The digital and / or quantum subject matter and implementations of digital functional operations and quantum operations described in this specification may be implemented in digital electronic circuitry, suitable quantum circuitry (or more generally, a quantum computing system), in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0106] The embodiments of the digital and / or quantum themes described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or for controlling the operation of the data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more quantum bits, or a combination of one or more of them. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal that is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing device) capable of encoding digital and / or quantum information.

[0107] The terms "quantum information" and "quantum data" refer to information or data carried, held, or stored in a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, for example, having two or more levels. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational basis state is identified by a ground state and a first excited state, but it should be understood that other arrangements in which computational states are identified by higher-level excited states are also possible.

[0108] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all types of devices, equipment, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or further include a dedicated logic circuit (e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit)) or a quantum simulator (i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system). In particular, a quantum simulator is a dedicated quantum computer that does not have the ability to perform general-purpose quantum computations. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, for example, code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

[0109] A digital computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and may be converted into a suitable quantum programming language, or may be written in a quantum programming language such as QCL or Quipper.

[0110] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program may be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple collaborative files (e.g., files storing some portions of one or more modules, subroutines, or codes). A digital and / or quantum computer program may be deployed to execute on a digital computer or a quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can use quantum systems (e.g., qubits) to transmit quantum data. Generally speaking, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0111] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers operating in conjunction with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by a dedicated logic circuit (e.g., an FPGA or ASIC) or a quantum simulator, and the apparatus can also be implemented as a dedicated logic circuit or a quantum simulator, or by a combination of a dedicated logic circuit or a quantum simulator and one or more programmed digital and / or quantum computers.

[0112] When a system of one or more digital and / or quantum computers is "configured to" perform a particular operation or action, it means that the system has software, firmware, hardware, or a combination thereof installed thereon that, when in operation, causes the system to perform the operation or action. When one or more digital and / or quantum computer programs are configured to perform a particular operation or action, it means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by the quantum computing device, cause the device to perform the operation or action.

[0113] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or special-purpose digital and / or quantum processor, or both, or any other kind of central digital and / or quantum processing unit. In general, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system (e.g., photons) suitable for transmitting quantum data, or a combination thereof.

[0114] The essential elements of a digital and / or quantum computer are a central processing unit for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory may be supplemented by or incorporated into specialized logic circuits or quantum simulators. Generally speaking, a digital and / or quantum computer will also include one or more mass storage devices (e.g., magnetic, magneto-optical disks, optical disks) for storing digital and / or quantum data, or a quantum system suitable for storing quantum information, or be operatively coupled to receive digital and / or quantum data therefrom, transmit digital and / or quantum data thereto, or both. However, a digital and / or quantum computer need not necessarily have such devices.

[0115] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media, and memory devices, including, for example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device that can store quantum data for a long time with high fidelity and efficiency, such as a light-matter interface, where light is used for transmission and matter is used to store and maintain quantum characteristics of quantum data, such as superposition or quantum coherence.

[0116] Control of the various systems or portions of the systems described in this specification may be implemented in a digital and / or quantum computer program product comprising instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions of the systems described in this specification may be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0117] Although this specification contains many specific implementation details, these details should not be interpreted as limiting the scope of the claimed invention, but should be interpreted as descriptions of features that may be specific to a particular implementation. Certain features described in this specification in the context of a separate implementation may also be implemented in a single implementation in combination. On the contrary, various features described in the context of a single implementation may also be implemented individually in multiple implementations or in any suitable sub-combination. In addition, although features may be described as working in certain combinations and even initially claimed as such, in some cases, one or more features from the claimed combination may be excluded from the combination, and the claimed combination may be directed to a sub-combination or a variant of the sub-combination.

[0118] Similarly, although operations are depicted in a particular order in the accompanying drawings, this should not be understood as requiring that such operations be performed in the particular order shown or in a sequential order, or that all illustrated operations be performed to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Additionally, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments. Instead, it should be understood that the described program components and systems may generally be integrated together in a single software product or packaged into multiple software products.

[0119] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method performed by a quantum computing device, the method comprising: Applying a quantum circuit to a qubit system included in the quantum computing device, wherein the quantum circuit implements a time-evolved Trotter step, comprising: applying a multilayer quantum logic gate to the qubit system, wherein each application of a layer in the multilayer evolves the qubit system under a unitary operator generated by a corresponding set of interaction terms in a qubit Hamiltonian representing the Fermi system, and each layer in the multilayer quantum logic gate comprises a rotation operation having the same rotation angle, wherein applying the layers of the quantum logic gate to the qubit system comprises assigning Hamming weights corresponding to the rotations in the layers of the quantum logic gate; and The evolving qubit system is measured to determine the energy of the Fermi system.

2. The method according to claim 1, wherein The qubit Hamiltonian is expressed in a plane wave dual basis, and each interaction term includes a corresponding translation-invariant coefficient.

3. The method according to claim 2, wherein: The values ​​of the translation-invariant coefficients of each set of interaction terms in the qubit Hamiltonian are constant.

4. The method according to claim 1, wherein The qubit Hamiltonian further comprises one or more kinetic energy terms, and wherein the time-evolving Trotter step comprises a split Trotter step that separates the evolution of the qubit system under the kinetic energy term and the interaction term.

5. The method according to claim 4, wherein Applying the quantum circuit includes applying a basis change circuit and an inverse of the basis change circuit to terms in the qubit Hamiltonian, wherein the basis change circuit includes a fast Fermi Fourier transform operation or a Givens rotation.

6. The method of claim 1 , further comprising generating the multi-layer quantum logic gate, generating the multi-layer quantum logic gate comprising, for a system of N qubits, where N is an even number: The N qubits are arranged as a two-dimensional grid of qubits, wherein The position of each qubit in the grid is represented by the corresponding position vector p; For each index vector s: For each position vector p on all N qubits, define a set of qubit pairs (p, p+s); Based on a defined set of qubit pairs, two layers of qubit interactions are generated, where each qubit in each layer interacts with another qubit.

7. The method according to claim 6, wherein: Each layer in the multi-layer quantum logic gate includes N / 2 equiangular rotation operations.

8. The method according to claim 6, wherein: Defining a set of qubit pairs (p, p+s) for each qubit p includes evaluating p+s modulo the grid length.

9. The method of claim 1 , further comprising generating the multi-layer quantum logic gate, generating the multi-layer quantum logic gate comprising, for a system of N qubits, where N is an odd number: The N qubits are arranged as a two-dimensional grid of qubits, wherein The position of each qubit in the grid is represented by the corresponding position vector p; For each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p on all N qubits, wherein a qubit is not included in the defined set of pairs; Generate three-layer qubit interactions based on a defined set of qubit pairs, including: The first layer, where each qubit interacts with every other qubit, a second layer, where each qubit interacts with every other qubit, and The third layer consists of a single rotation operation.

10. The method according to claim 9, wherein: The first and second layers of the quantum logic gate each include (N-1) / 2 equiangular rotation operations.

11. A device comprising: Quantum computing hardware, including: qubit systems; and a control device configured to apply a quantum circuit of quantum logic gates to qubits included in the qubit system and measure the qubits included in the qubit system; The device is configured to perform operations, including: Applying a quantum circuit to a qubit system included in the quantum computing device, wherein the quantum circuit implements a time-evolved Trotter step, comprising: applying a multilayer quantum logic gate to the qubit system, wherein each application of a layer in the multilayer evolves the qubit system under a unitary operator generated by a corresponding set of interaction terms in a qubit Hamiltonian representing the Fermi system, and each layer in the multilayer quantum logic gate comprises a rotation operation having the same rotation angle, wherein applying the layers of the quantum logic gate to the qubit system comprises assigning Hamming weights corresponding to the rotations in the layers of the quantum logic gate; and The evolving qubit system is measured to determine the energy of the Fermi system.

12. The device according to claim 11, wherein The qubit Hamiltonian is expressed in a plane wave dual basis, and each interaction term includes a corresponding translation-invariant coefficient.

13. The device according to claim 12, wherein The values ​​of the translation-invariant coefficients of each set of interaction terms in the qubit Hamiltonian are constant.

14. The device according to claim 11, wherein The qubit Hamiltonian further comprises one or more kinetic energy terms, and wherein the time-evolving Trotter step comprises a split Trotter step that separates the evolution of the qubit system under the kinetic energy term and the interaction term.

15. The device according to claim 14, wherein Applying the quantum circuit includes applying a basis change circuit and an inverse of the basis change circuit to terms in the qubit Hamiltonian, wherein the basis change circuit includes a fast Fermi Fourier transform operation or a Givens rotation.

16. The apparatus of claim 11, wherein the operations further comprise generating the multi-layer quantum logic gate, generating the multi-layer quantum logic gate comprising, for a system of N qubits, where N is an even number: The N qubits are arranged as a two-dimensional grid of qubits, wherein The position of each qubit in the grid is represented by the corresponding position vector p; For each index vector s: For each position vector p on all N qubits, define a set of qubit pairs (p, p+s); Based on a defined set of qubit pairs, two layers of qubit interactions are generated, where each qubit in each layer interacts with another qubit.

17. The device according to claim 16, wherein Each layer in the multi-layer quantum logic gate includes N / 2 equiangular rotation operations.

18. The device according to claim 16, wherein Defining a set of qubit pairs (p, p+s) for each qubit p includes evaluating p+s modulo the grid length.

19. The apparatus of claim 11, wherein the operations further comprise generating the multi-layer quantum logic gate, generating the multi-layer quantum logic gate comprising, for a system of N qubits, where N is an odd number: The N qubits are arranged as a two-dimensional grid of qubits, wherein The position of each qubit in the grid is represented by the corresponding position vector p; For each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p on all N qubits, wherein a qubit is not included in the defined set of pairs; Generate three-layer qubit interactions based on a defined set of qubit pairs, including: The first layer, where each qubit interacts with every other qubit, a second layer, where each qubit interacts with every other qubit, and The third layer consists of a single rotation operation.

20. The device according to claim 19, wherein The first and second layers of the quantum logic gate each include (N-1) / 2 equiangular rotation operations.

21. A method for determining a property of a physical system, the method being performed by a system comprising one or more classical processors and quantum hardware, the quantum hardware comprising: A system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and to measure the system of interacting qubits, the method comprising: Generating a multi-layer quantum logic gate, wherein each layer in the multi-layer quantum logic gate includes a rotation operation having a same rotation angle, the generating comprising: For a system of N even number of qubits and for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p over all N qubits, where the position of each qubit is represented by the position vector p, and generating two layers of qubit interactions based on the defined set of qubit pairs, where each qubit in each layer interacts with another qubit; or for a system of N odd number of qubits and for each index vector s: defining a set of qubit pairs (p, p+s) for each position vector p over all N qubits, wherein one qubit is not included in the defined set of pairs, and generating three layers of qubit interactions based on the defined set of qubit pairs, the three layers comprising: i) a first layer in which each qubit interacts with another qubit, ii) a second layer in which each qubit interacts with another qubit, and iii) a third layer comprising a single rotation operation; A system for applying the multi-layer quantum logic gate to qubits; and A system of qubits is measured by the quantum hardware to determine one or more properties of the physical system.

22. The method according to claim 21, wherein Each layer in the multi-layer quantum logic gate includes N / 2 equiangular rotation operations.

23. The method according to claim 21, wherein The first and second layers of the quantum logic gate each include (N-1) / 2 equiangular rotation operations.

24. The method according to claim 21, wherein The physical system is described by a corresponding Hamiltonian, and wherein the method further comprises converting, by the one or more classical processors, the Hamiltonian describing the physical system into a qubit Hamiltonian describing a system of qubits, wherein the qubit Hamiltonian comprises a plurality of two-qubit interaction terms, each two-qubit interaction term comprising a corresponding translation-invariant two-qubit interaction term coefficient.

25. The method according to claim 24, wherein Applying the multi-layer quantum logic gate to a system of qubits evolves the system of qubits under a unitary operator generated by the plurality of two-qubit interaction terms.

26. The method according to claim 25, wherein The system for applying the multi-layer quantum logic gate to a qubit further includes, in each layer of the multi-layer, applying Hamming weight phasing to rotations in the layer, wherein applying the Hamming weight phasing comprises: determining Hamming weights for the qubits in the layer; and A rotation is applied to the qubit based on the Hamming weight.

27. The method according to claim 24, wherein The qubit Hamiltonian further includes a kinetic energy term and a single-qubit potential energy term, and wherein the method further includes evolving the system of qubits under a unitary operator generated by the kinetic energy term and a unitary operator generated by the single-qubit potential energy term.

28. The method according to claim 25 or 27, wherein Systems for evolving qubits under a unitary operator generated by the plurality of two-qubit interaction terms, kinetic energy terms, or single-qubit potential energy terms include implementing time evolution or implementing directionally controlled evolution on an auxiliary bit.

29. The method according to claim 28, wherein Evolving the system of qubits under a unitary operator generated by the plurality of two-qubit interaction terms, kinetic energy terms, or single-qubit potential energy terms includes evolving the system of qubits in a split-operator Trotter step.

30. The method according to claim 29, wherein The splitting operator Trotter step is one of a plurality of splitting operator Trotter steps that produces a minimum Trotter error.

31. The method according to claim 21, wherein The Hamiltonian describing the physical system includes the electronic structure Hamiltonian.

32. The method according to claim 21, wherein The physical system comprises a chemical substance or material, and wherein determining a property of the physical system comprises determining a property of the chemical substance or material.

33. An apparatus for determining a property of a physical system, the apparatus comprising: One or more classic processors; Quantum hardware includes: A system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and to measure the system of interacting qubits; The apparatus is configured to perform the method according to any one of the preceding claims 21-32.

34. A method for determining a property of a physical system, the method being performed by a system comprising one or more classical processors and quantum hardware, the quantum hardware comprising: A system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and to measure the system of interacting qubits, the method comprising: generating a multi-layer quantum logic gate, wherein each layer in the multi-layer quantum logic gate evolves the system of interacting qubits under a corresponding unitary operator generated by a corresponding set of two-qubit interaction terms included in a qubit Hamiltonian describing the physical system and includes rotation operations having the same rotation angle; combining rotation operations within each layer of the multi-layer quantum logic gate using Hamming weight phasing to generate a plurality of updated layers of the quantum logic gate; Systems that apply multiple update layers of quantum logic gates to qubits; and A system of qubits is measured by the quantum hardware to determine one or more properties of the physical system.

35. An apparatus for determining a property of a physical system, the apparatus comprising: One or more classic processors; Quantum hardware, including: A system of interacting qubits, wherein the interactions between the qubits are controllable; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and to measure the system of interacting qubits; Wherein, the apparatus is configured to perform the method according to claim 34.

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