Quantum computing system and method for noise mitigation

Abstract

Mid-circuit measurements and feed-forward conditioning are integrated as computational primitives within a quantum channel that is executed on the qubits of a quantum computing system. This provides r

Description

Technical Field The present disclosure relates to noise mitigation within a quantum computing system, such as a hybrid computing system including a combination of a classical binary computer coupled to a quantum computer. Moreover, the present disclosure relates to a method for performing computations within a quantum computing system. Furthermore, the present disclosure relates to software products recorded on machine-readable media, wherein the software products include program code which is executable to control generation of quantum circuits and operation of a quantum computing system. Background The main obstacle to large-scale quantum computing is the imperfect realization of quantum operations [1, 2] and a longstanding problem in the development of quantum computers has been how to maintain quantum performance, such as quantum coherence, in the presence of noise. Various sources of noise, such as heat or electromagnetic radiation, can cause unwanted disturbances to the devices carrying out quantum computations, effecting control of qubits and interfering with reliable and accurate execution of quantum operations as well as causing information within the signal output to be lost. Quantum computers have recently become available as noisy intermediate-scale quantum (NISQ) devices, and quantum computers have also been emulated using classical computers, but there is a need to improve reliability and scalability. The presence of noise and the current lack of complete fault-tolerance causes significant limitations in the types of algorithms that are executable on currently-available NISQ devices, where considerations such as the circuit depth or the available resources (e.g., limited number of qubits and a lack of qubit fidelity) need to be taken into account. Therefore, mitigation of the problems of noise is a significant technical problem associated with NISQ devices. Various methods have been proposed in published scientific literature that account for errors that arise when performing computation on NISQ devices, such as error detection and error correction methods, alongside efforts to improve qubit fidelity. However, due to limitations in the scalability and accuracy of current NISQ devices, existing quantum computers can only implement quantum error correction protocols on few logical qubits with limited circuit depth. In the presence of noise, the signal output by a quantum circuit is generally exponentially attenuated with the number of gates [3-7], causing exponentially increasing overheads which has limited the prospects for utility of currently available quantum computers. Summary The present disclosure provides improved methods and systems for mitigation of the effects of noise when performing computations using a quantum computing system. Described below are quantum computing systems and computation methods in which a quantum channel describing the evolution of a physical system incorporates one or more mid-circuit measurements with at least one subsequent computation step which is conditional on the measurement outcome, wherein the mid-circuit measurements and conditional computation steps are integral to a quantum channel that outputs a mixed state. The result of this integration is that the quantum circuit that prepares the mixed state is naturally noise-resilient and avoids complete decoherence. The quantum computing systems and computation methods can provide a non-zero signal output that contains useful information about the evolution of a physical system, even for a large circuit depth or after an arbitrarily large number of repeats in the presence of noise. A first computer-implemented computation method comprises: generating a quantum channel, including operations to be performed with respect to quantum states, to represent the evolution of a physical system, wherein the quantum channel comprises quantum circuits adapted to perform at least a first computation step on a set of qubits (or qudits) including at least one anciIla qubit (or qudit), followed by at least one mid-circuit state measurement performed on the at least one ancilla qubit (or qudit), and to perform at least a second computation step that is conditional on a result of the mid-circuit state measurement, to generate a mixed state output of the quantum channel; and executing the at least one quantum channel on a set of qubits or qudits of a quantum computing apparatus, including performing at least a first computation step followed by performing the at least one mid-circuit measurement on an output of at least one ancilla qubit (or qudit) of the set of qubits (or qudits), and performing the at least one conditional computation step on a subset of the qubits (or qudits) other than the measured ancilla qubit (or qudit), thereby to generate a mixed state output containing information relating to the physical system. Quantum channels are used to describe the evolution of physical or computational quantum systems as they interact with their surroundings or undergo various operations. Quantum channels represent the transformation of quantum states due to those interactions (with the environment or intentional operations within a quantum system) using mathematical operations. The quantum channel execution may be repeated multiple times to stabilize a mixed state output on the qubits of the quantum computing system. The method would output a mixed state (obtained as a 'fixed point' or equilibrium state of the quantum channel) even in the absence of noise, but is especially useful for mitigating the effects of noise as the fixed point varies only smoothly with noise strength such that the mixed state that is produced is not a completely depolarised state. Hereafter in this patent specification, any reference to qubits should be interpreted as a reference to qubits or qudits, as the invention is equally applicable to both. At least one mid-circuit measurement and at least one computation step that is conditional on the measurement result are preferably implemented as integral computational primitives within the quantum channel itself. This integration builds noise-resilience into quantum computations. The output of execution of the at least one quantum channel on a quantum computer is a mixed state output, which can provide information relating to the physical system even in the presence of noise, partial decoherence, or imperfect measurements. A mixed state is a statistical ensemble of quantum states with a matrix of probabilities of finding the system in different pure states, for example described by density matrices. Methods according to the invention can generate a mixed state output of the quantum channel on the system qubits that cannot be produced by non-unitary operations (without incurring a runtime that is exponentially large in the number of qubits) and improves upon purely unitary unconditional operations, but the operations that are conditionally performed on the system qubits based on the ancilla qubit measurement are preferably implemented as unitary, non-destructive operations - i.e. preserving the probability of different outcomes. The computer system may comprise a classical (digital) computing apparatus and a quantum computing apparatus, with the generation of the quantum channel performed on the classical computing apparatus and the execution of the quantum channel performed on the quantum computing apparatus. A method of computation preferably comprises providing a representation of a physical system as an input to a computer system, for use in generation of a quantum channel. The representation of the physical system includes at least one operator to perform operations with respect to a quantum state of the represented physical quantum system. This representation of the physical system may be a Hamiltonian, for example for mathematically representing the energy states and interactions of a physical quantum system, or may be a mathematical representation of the energy states of a classical system. A Hamiltonian can be used to generate one or more quantum circuits for acting on a set of qubits of the quantum computer system. Initialisation of a quantum state followed by execution of the quantum circuits on the qubits and measurement of qubit outputs provides information about the physical system represented by the Hamiltonian. A quantum channel with integral mid-circuit measurements, and integral conditional execution of one or more computational steps within the quantum channel, differs from known quantum error mitigation techniques that rely on separate error-checking subroutines that do not perform any computation beyond their error checking. It is known to provide subroutines for mid-circuit measurement that are used to detect errors and to decide whether to restart the entire computation from the beginning (i.e. if certain errors are a stopping condition), where the final output after postselection is a pure state. Quantum-error-corrected circuits would output a pure state in the absence of noise. However, the present invention differs by its integration of midcircuit measurements performed on one or more ancillas with conditional performance of the subsequent computation also implemented as an integral part of the quantum channel itself, and by outputting a mixed state. The conditional computation steps can be implemented as computational primitives within the quantum channel, in contrast to separate error checking subroutines. The result is an integral noise-resilience that avoids some of the limitations which arise when implementing dedicated quantum error detection / correction subroutines on NISQ devices. A method according to the present invention provides a noise-resilient way of preparing quantum states, as the output from the described quantum circuits, in the form of a mixed state. This state preparation could be used for a wide range of subsequent computations, performed on the same or different qubits. One of the potential applications of the present invention is for initialisation of qubits to a specific mixed quantum state, in advance of a subsequent quantum computing operation. However, the present invention's combination of mid-circuit measurement and "feed forward" performance of operations conditional on this measurement may be integrated within a wide range of computations, where it is used to add noise resilience to that computation. The preparation of this mixed state is resilient to noise in the sense that noise will only smoothly modify the output signal even for large circuit depth (whereas standard unitary circuits always end up completely depolarized at large circuit depth in presence of noise, even with error correction routines which only delay the depolarizing). Contrary to previously known codes, implementations of the present invention can output a non-zero signal containing useful information about the transformation / evolution of a physical system, even after an arbitrarily large number of repeats in the presence of noise. For some implementations, the invention provides this noise resilient behaviour with a low qubit overhead, which is advantageous when executing on NISQ devices. For example, the invention is implementable with an overhead of only one ancilla qubit, or alternatively using multiple ancillas, in combination with the qubits that are used to perform the complete computation. This low qubit overhead is possible when protecting a specific outcome within the quantum channel, contrary to known error-correction routines which are intended to be applicable to a variety of different circuits. In an embodiment, execution of the quantum channel comprises the following steps. Firstly, a classical random variable is generated and the qubits are initially prepared in a state independent of any previous calculations or measurement outcomes. An operator that depends on the random variable is applied to a set of qubits of a quantum computing apparatus including one or more ancilla qubits. This operator can be a unitary operator that entangles the one or more a nci I la qubits with the other qubits of the set, or another operation that couples (i.e. establishes a correlation between) the anciIlas and the other qubits. For example, this coupling may use a time evolution operator eA{i H X} applied to a set of qubits including the one or more ancillas, where H is a Hamiltonian applied to the system, and X is a Pauli matrix applied to the ancilla, which ancilla is prepared in state |0) . Measurements can then be carried out on the one or more ancillas, with the results of the measurement used for conditional control of one or more subsequent computation steps performed on a subset of the qubits excluding the measured ancilla(s). For example, an operation may be performed on L qubits and one ancilla qubit, where L is a number of qubits used for a computation; a mid-circuit measurement may be performed on the one ancilla qubit; and then the computation may be continued on the L qubits conditional on the result of the mid-circuit measurement. The mixed state output of the first iteration can then be used as an initial state for subsequent iterations of the execution of the quantum channel, repeating until a relatively stable mixed state output is achieved or repeating an arbitrarily large number of times. The computation method may be applied, with a low qubit overhead, in the computation of dynamical correlations with an assumption that all outcomes are equally likely (referred to as dynamical correlations at "infinite temperature"). The invention is also applicable for performing computation of finite temperature expectation values for a Hamiltonian, if we use multiple ancillas. A first quantum computing system comprises: a first computing apparatus configured to generate a quantum channel for performing a sequence of computational steps for computing information relating to a physical system, the quantum channel comprising quantum circuits adapted to perform at least a first computation on a set of qubits including at least one ancilla qubit, and wherein the quantum channel comprises at least one mid-circuit state measurement performed on the at least one ancilla, and the quantum channel comprises at least one computation step that is conditional on a result of the mid-circuit state measurement, and wherein the quantum channel is adapted to generate a mixed state output; and a quantum computing apparatus comprising a set of qubits for executing the quantum channel, including performing at least a first computation step followed by performing the at least one mid-circuit measurement on an output of at least one ancilla qubit of the set of qubits, and performing the at least one conditional computation step on a subset of the qubits other than the measured ancilla qubit, thereby to generate a mixed state output containing information relating to the physical system. The quantum channel may be implemented with only a single ancilla qubit, or with multiple qubits. The first computing apparatus may be a classical digital computing apparatus. Brief Description of Figures Described below are methods, systems, apparatus and computer programs for implementing the present invention, provided by way of example only, with reference to the accompanying drawings in which: Figure 1 is a schematic representation of an example quantum computing system, including a classical computing apparatus coupled to a quantum computing apparatus and arranged to cooperate to execute computational tasks, involving the generation and execution of a quantum channel; Figure 2 shows a sequence of steps of an example method implementing the invention; Figure 3 represents a quantum channel being executed on a set of qubits including: premeasurement steps, followed by mid-circuit measurement of an ancilla qubit, followed by feedforward conditional performance of computations on system qubits other than the ancilla qubit; Figure 4 shows steps of a method implementing some of the steps of Figure 2; Figure 5 represents the performance of operations of a quantum channel using a single ancilla qubit; Figure 6 represents repeated performance of operations of a quantum channel using a single ancilla; Figure 7 represents the performance of operations of a quantum channel using multiple ancilla qubits; Figure 8 is a graphical representation of the noise-mitigation effects of a quantum channel according to the invention; and Figures 9 and 10 are graphical repreentations of dynamical correlation results for example quantum channels according to the invention. Detailed Description The inventors of the present invention have determined that noise-resilience can be made part of the algorithmic design of computational algorithms for running on a quantum computer system, by incorporating mid-circuit measurements and feed-forward conditional computation as integral computational primitives within a quantum channel-i.e. within a quantum computing circuit that computes information about a physical system. The channel is prepared to generate a mixed state output. The mid-circuit measurements and at least one subsequent computational step that is conditional on the measurement outcomes are implemented as integral computational tools (rather than error correcting subroutines), making the computations naturally noise-resilient in the sense that, even in the limit of an infinitely deep circuit, there remains a non-zero signal that contains information on the task fulfilled by the algorithm. Figure 1 is a schematic illustration of a hybrid quantum computing system 10 including a classical computing apparatus 20 coupled to a quantum computing apparatus 30. The classical computing apparatus and the quantum computing apparatus are configured to operate in tandem to execute computational tasks involving the generation and execution of a noise resilient quantum channel. Specific quantum channels for implementing the present invention are described below. As shown in Figure 2 and Figure 3, for a single execution of the channel, the steps of a method for implementing the invention can be represented as a simple sequence of steps. The method starts with receipt 100 of data such as an input Hamiltonian or other information about a quantum system for which a computation is required. A quantum channel is generated 110 by a quantum circuit builder 22 running on the classical computing apparatus 20 of the hybrid quantum-classical system 10. This quantum channel comprises quantum circuits for performing at least a first computation on a set of qubits 32,34 of the quantum computing apparatus 30, which we will refer to as a number L of system qubits 32 and an ancilla (or auxiliary) qubit 34. The quantum channel is generated to include, as computational primitives integral to the quantum channel: a mid-circuit measurement 132 carried out on the ancilla qubit 34; and at least one subsequent computation 142 to be performed on the L system qubits which is conditional on the outcome of the ancilla measurement. Following generation 110 of the quantum channel, the generated quantum circuits are executed on the qubits 32,34 of the quantum computing apparatus 30, including performing 120,126 at least a first computation on the L qubits and ancilla qubit, and then performing 130,132 a mid-circuit measurement on the ancilla qubit. Additionally, the subsequent computation is performed 140,142 on the L qubits, conditional on the outcome of the measurement result of the ancilla qubit. The result of the computation is prepared as a mixed state output. This sequence of steps can then be repeated, using the mixed state output of the first iteration as an input for the next iteration. Some steps of the method are shown in more detail in Figure 4, byway of example. Steps 110 and 120 of Figure 2 can be implemented using steps 122, 124 and 126 of Figure 4; and step 130 of Figure 2 can be implemented with steps 132 and 134 of Figure 4. As shown in Figure 4, for a representation of a physical system such as a Hamiltonian, a quantum channel may be defined as an operation or set of operations to be performed in relation to the states of a physical quantum system. This can be implemented by allocating 122 a Pauli matrix to each of L qubit sites, where L is a number corresponding to the set of available qubits. We prepare 124 an a nci I la qubit in a first initialized state, |0)a, and then we perform 126 a first computation operation on each of the L qubits and the ancilla. Next, we measure 132 the output of the ancilla qubit. One or more subsequent computations 142 of the quantum channel are made conditional on the determined result 134 of that measurement-i.e. only if the measurement outcome is a specific result such as 0, we apply 142 the respective Pauli matrix at each qubit site. The dashed lines returning to step 122 are to represent the possibility of repeated execution of the quantum channel. This processing is represented by Figures 5, 6 and 7 with reference to certain examples of a quantum channel as defined below and shown schematically in Figure 3. We start with an initial density matrix 200 of the L qubits, and a single ancilla qubit 210, as shown in Figure 5. Execution of the quantum channel 300 outputs a new density matrix 200' and one observable ancilla qubit state 210'. Figure 6 is similar to Figure 5, but with the quantum channel repeated n times. Figure 7 is also similar to Figure 5, but with a plurality of ancilla qubits. These options are described below. The invention has wide applicability but, as a first specific example, we introduce a simple "shuffling" quantum channel based on mid-circuit measurements and feed-forward that outputs a non-zero signal when repeated an arbitrary number of times, even in the presence of noise. Crucially, this non-zero signal also carries non-trivial, physically interesting information: We show that this quantum channel can be used to compute finite temperature expectation values for any Hamiltonian H, as well as dynamical correlations at "infinite temperature" tr [Se,Ht5e-lHt] where S is any Pauli string. The condition of "infinite temperature" is a conceptual condition in which all states are assumed to be equally likely. The inventors have also shown that the described method can be implemented to prepare certain highly quantum correlated density matrices in a noise-resilient way. The use of mid-circuit measurements to perform non-unitary operations is known, for example as a means of post-selecting particular measurement outcomes [14-23], to prepare specific, analytically tractable quantum states more efficiently [24-29], or on top of another algorithm to mitigate the effect of noise [30-32], Some algorithms use a weak form of feed-forward where measurement outcomes are used as a stopping criterion or to partially reset the state [33-38], The present invention differs from these known techniques by the use of mid-circuit measurement and feed-forward as computational primitives to obtain non-trivial physical properties which endow the computation with a natural noise-limiting property. This noiselimiting behaviour has been demonstrated analytically with specific noise models, numerically with noisy simulations, and experimentally on ion-trap quantum computers from Quantinuum™. The shuffling quantum channel A first quantum channel is described below by way of example, with reference to Figures 3 and 4. The effect of the channel on expectation values is given by Equation (2) below. The inventors have determined that a quantum channel which implements mid-circuit measurements with feed forward conditionality within the channel can be used to compute dynamical correlations at infinite temperature. An unusual property of this channel is that when repeated several times in presence of noise, it still outputs meaningful non-zero signal. For a generic Hamiltonian H on L qubits, we define the quantum channel Shuffle(H) by the three following steps. 1. Pick a Pauli matrix at each site, i.e. M1,...,MlE{I,X,Y,Z} uniformly at random. 2. Prepare an ancilla in state |0)a.Onthe L qubits and the ancilla, apply einM±„MLxa where we defined = (1) 3. Measure the ancilla. Only if the measurement outcome is 0, apply on the L qubits. Let us denote p the initial density matrix of the L qubits, and p' the new density matrix after one run of Shuffle(H). We are going to show that for any Pauli string operator S = M\... M'L with M'i G { / , X, Y, Z}, we have: tr [Sp'] = Atr [Sp] + B , (2) with , _ tr [Ssin(H)Ssin(H)] D _ tr [Scos2( / f)] , . A ~' B ~ ' (3) Moreover, the probability p of measuring 0 in the ancilla is tr[cos2(H)l P= 2l ■ (4) Proof From step 1 of Shuffle(H) above, we fix some Pauli matrices at each site M±, G {I,X, Y, Z}. We will denote M = and M the set of all the 4L Pauli strings. The operator U = eiHMxa js diagonal in the eigenbasis of WM, with matrix elements Un that are operators on the ancilla and that read II = (cosEn isinEn\ n Vising cosEn / ' with En the corresponding eigenvalue of HM. By preparingthe ancilla in the state |0), applying U and measuring the ancilla, we act on the L qubits with the operator (0|t / |0) = cos(f / M) if we measure 0, and with = isin(HM) if we measure 1. It follows that the density matrix p'M that we obtain after the steps (2) and (3), assuming ...,ML have been picked at step (1), is Pm = sin(WM)psin(WM) + Mcos(HM)pcos(HM)M = sin(WM)psin(HM) + cos(H)MpMcos(H) . Taking into account all the 4L possibilities of M1,...,ML at step 1 of Shuffle(H), the density matrix p' after one run of Shuffle(H) is the sum of all the p'M's divided by 4L. We now note that on the second term of p'M, this sum precisely implements a completely depolarizing channel

[13] , Namely, for any density matrix p we have 1 1 MpM = ^. (7) It follows p’ = (8) We note that the feed-forward in step 3 of Shuffle(H) is important to obtain a non-trivial density matrix, following the mid-circuit measurement. Had we always applied M on the qubits independently of the measurement outcome of step 2 of Shuffle(H), the density matrix p' would have been the completely depolarized state 1 / 2^. Let us now consider a given Pauli string operator M' 6 M, and denote the expectation values m = tr [M'p], m' = tr [M'p']. (9) Multiplying (8) by M' and taking the trace, we make appear in the sum the terms tr [M'sin(HM)psin(HM)]. Decomposing p in the basis of Pauli strings P =TiM"em (10) we write tr [M'sin(HM)psin(HM)] M .. = cM"tr [MM,Msin(H)MM"Msin( / 7)]. ' ' Let us perform first the sum over Mi G { / , X, Y,Z] appearing in (8). We note that MiM^Mi is either M / (if Mi = I, or M / = I, or Mt = M / ), or —M / (in the remaining cases). If M / M", then at least one of the two is not 1, say M / If Mi" = I, then one sees that one gets an amplitude tr [M'sin(7 / )M"sin(H)] with a + sign when Mt = / and Mi = M / , and with a — sign in the two remaining cases. Hence this term vanishes once summed over Mi- If M / = I and M"i I, the same reasoning applies. If both M^.M” I, one sees that one gets a + sign when Mt = I and Mt #= M / .M / ', and a — sign in the two remaining cases. Hence this term also vanishes once summed over Mt. It follows that once we sum over all the Mif only the terms where Mt = Mi" give a non-zero contribution. This yields Smem tr [MM'Msin(H)MM"Msin(H)] = 4Ltr [M'sin(H)M"sin(H)] Hf=i ■ (12) Noting that we have m = cM, 2L, we obtain thus tr [M,sin(H)M'sin(H)] tr [M'cos2( / f)] 2L 2l (13) which is precisely (2). To compute the probability p of measuring 0 in the ancilla, we note that if has been picked at step 1 of Shuffle(H), then this probability pM is pM = tr [pcos2(HM)] . (14) Summing over M1(..., ML, this yields p= (15) Using in this equation the relation (7) for the density matrix we obtain (4). Dynamical correlations at infinite temperature Let us show how the shuffling quantum channel can be used to compute dynamical correlations of Pauli strings S = Mx. ,.ML at infinite temperature, defined as ■ (16) These quantities are relevant for transport properties and have attracted particular interest recently [39-48], To compute this quantity with the shuffling quantum channel, we write _ , . _ tr [Scos( / ft)Scos(Ht)] tr [Ssin(Ht)Ssin(Ht)] . . ^s\y) - । ■ (1 / ) Let us consider p an arbitrary initial density matrix for which we know tr [pS]. We then apply on p either Shuffle( / / t) with probability 1 / 2, or Shuffle(Ht+-) with probability 1 / 2. We measure the Pauli string S and denote tr [p'S] the average outcome. Using formula (2), we have thus tr [p S] = —tr [pS] + —, (18) with AX,BX the coefficients (3) correspondingtothe Hamiltonian Ht + x. From (3) we thus find tr [p'S] = . (19) Hence, one can directly compute the value of C$(t) provided we chose p such that tr [pS] + 0. There are many other similar ways of extracting Cs(t) from this quantum channel

[49] , This algorithm for dynamical correlations at infinite temperature presents advantages compared to previously existing techniques. Computing the trace over the Hilbert space through a purification requires to double the number of qubits, whereas here there is only one ancilla qubit overhead. Implementing the trace through a Haar random state preparation

[49] has a gate overhead to prepare the Haar random state, and also a shot overhead because it computes the square of the dynamical correlations and can only treat one site at a time, which increases considerably the variance. Finally, for S single Pauli matrices, evaluating the trace as a sum over product states in the basis of S can have a large statistical sampling overhead if dynamical correlations are small in this basis, whereas our algorithm is basis-independent. Dynamical correlations at different lattice sites such as tr [Z^+1(t)Z1(0)] / 2L can be obtained as well with the following modification of the channel. Just after applying elHMi-MLXa in the second step of the channel, we apply the operator that swaps qubits 1 and ^ + 1 conditioned to Xa being —1. Density matrix preparation The density matrix constructed previously in (19) contains information about the physics of the system through Cs(t), but also on the initial density matrix through tr [pS]. This can be problematic for strings S involving different Pauli matrices at different sites, since the preparation of p such that tr [pS] #= 0 for multiple such S would become non-trivial. The shuffling quantum channel through the relation (2) actually enables one to prepare density matrices with high quantum correlations independently of the initial density matrix. With reference to figures 3 and 5, let us repeat n times the channel Shuffle(H) on an initial density matrix p0, denoting pn the resulting density matrix. Equation (2) yields a geometric series for any string S tr [Spn] = Antr [Sp0] + 7777 B , (20) with A,B given in (3). In particular, since |Z11 <1 (except for very specific choices of H,S where we would have |^4| = l)when n -* 00 we obtain lim tr[Spn] =-^-, (21) n^co i— a which does not depend on the initial density matrix. Moreover the convergence is exponentially fast in n with rate logl / |X|. This can be applied to the dynamical correlation algorithm in the following way. If at each round we apply either Shuffle (Ht) with probability 1 / 2 or Shuffle(Ht + with probability 1 / 2 we obtain lim tr [Spn] = -......................—, (22) n^co with exponential convergence in n with rate log2 / |Cs(t)| >log2 . This prepared density matrix p* satisfies for any Pauli string tr [p,S] >1 / 3. Remarkably, it is thus a highly long-range quantum correlated state whose expectation values contain physical information about the system studied. Noise-limiting behaviour: analytics As noted above, the algorithms based on the repetition of the shuffling quantum channel are to some extent resilient to noise. To give evidence for this claim, let us first consider a simple noise model where a global depolarizing channel is applied on the system qubits after each run of Shuffle(H). Namely, the density matrix p' after applying Shuffle(H) is sent to P'noisy = (1 — typ' + ■ Eq (2) thus becomes tr[Sp'noisy] = A'tr [Sp] + B', with A' = (1 — A) A and B' = (1 — A)B. It follows that after iterating the shuffling channel with the noise round, we obtain the following expectation values within the density matrix r , (l-A)B hnitr [Spno^n] - t _ ^A- Even if in this limit the circuit is infinitely deep and infinitely many depolarizing noise channels have been applied on the system, the system converges to a non-trivial density matrix that is not the completely depolarized mixed state. This is in contrast with noisy purely unitary circuits that would always converge to the completely mixed state in this large depth limit. We note that no measurement or reset is ever carried on the system qubits, but only on the anciIla. Let us explain why this noise-limiting behaviour is directly related to the use of mid-circuit measurements and feed-forward. A circuit involving only unitary operations and measurements without feed-forward can always be written as the composition of density matrix maps T(p) = UpU^ with U unitary or T(p') = ^mP^m with = Id. These maps both satisfy T(Id) = Id (also called "unital" maps), so the completely mixed state is always a fixed point of such circuits. However, with feed-forward these density matrix maps are in general T(p) — UmMmpMmUm with Um unitary, for which the identity is (generically) not a fixed point. This thus prevents the completely depolarized state to be reached in the limit of large number of gates. Together with the previous calculation with the global depolarizing noise channel, this yields the following intuitive explanation of the noise-resilience of our algorithm. The resulting density matrix after a large number of rounds converges to the fixed point of the quantum channel. Because of the presence of mid-circuit measurements and feed-forward, this fixed point cannot be (in general) the completely depolarized channel. Now, incorporating noise in the process (under any form) will only smoothly change the quantum channel, and so only smoothly change the fixed point, instead of sending it to the completely depolarized state. The noise-resilience of the prepared density matrix is thus built into the algorithm itself. The same calculation as above can be applied to slightly less simple noise channels. Let us consider now a noise model where a depolarizing channel with amplitude A is applied to each qubit after the application of the operator elHMX“ appearing in Shuffle(H). This noise model is equivalent to applying a X,Y,Z error each with probability A / 4 on the qubit, and applying no error with probability 1 — 3A / 4. We consider first the effect of applying X on qubit j after step (2) of Shuffle(H). This is equivalent to applying elH»Xa on XjpXj with M = XyM. Denoting p' the density matrix obtained after steps (1), (2) and (3) when an X error has occurred at site / , then in terms of the noiseless value p' in (8) one has exactly p' = Xjp'Xj . (23) We thus obtain tr [Spz] = tr \XjSXjp'} for a Pauli string S. Since Xj either commutes or anticommutes with S, we have tr [XjSXjp'] = +tr [Sp']. The + occurs with probability 1 — A / 2 and the - sign with probability A / 2. Hence the total noisy density matrix p'noisy after one round satisfies, taking into accountthe different possible errors: tr [Sp'noisy] = (1 - A)G4tr [Sp] + B) . (24) Denoting pnotsy,n the density matrix obtained after n noisy applications of Shuffle(H), the limiting value reached in the limit of an infinite number of applications is limtr [Spnoisvnl =............—.............—............., (25) n^co L rnoisy.nx ’ which is finite, even if an infinite number of gates has been applied. The same result as for the global depolarizing channel is recovered. Moreover in this case, the noise level A appearing here only refers to the probability of error per site. Noise-limiting behaviour: experiments and simulations We now present numerical and experimental evidence of the noise resilience of our algorithm. We consider the ID Ising model in a transverse field h = 1 and t = 0.1 on L = 8 sites with periodic boundary conditions Hlsing = — ZjZj+1 — Xj . (26) We first present evidence for the noise-resilience of the simple shuffling quantum channel Shuffle(H). We initialize the L qubits in |0) and measure |SF=i after n rounds of Shuffle(H). We chose this observable as or would have a zero expectation value at either small or large number of rounds. We show in Figure 8 the results of noiseless simulations, noisy simulations with the Hl emulator from Quantinuum™, and actual hardware implementation with the Hl-1 machine from Quantinuum for number of rounds n = 1 (185 two-qubit gates and 782 one-qubit gates) and n = 10 (1850 two-qubit gates and 7820 one-qubit gates). Figure 8 is a graphical demonstration of the noise-limiting property of the shuffling quantum channel. Expectation values ZZ = CZjZi+i) are plotted as a function of the number n of calls to the quantum channel referred to herein as Shuffle(H), with an example Hamiltonian H as disclosed in (33), and using 5 Trotter steps to implement elH. There are 1000 shots per circuit for simulations, and 500 shots per circuit for hardware. The errors bars indicate + one standard deviation. After compilation, there are 185n two-qubit gates and 782n one-qubit gates as a function of n. It is known that noise on quantum computers can be well accounted for, in a first approximation, by attenuating the signal with a factor equal to the product of the fidelities of the gates entering the circuit. We evaluate the effect of the noise by measuring the signal after one round n = 1 on the emulator with a large number of shots, and comparing it to the exact value. From this, we obtain an attenuation per round 1—2. We plot then in Figure 8 the effect of the noise assuming it can be modelled by an attenuation factor (1—2)” at round n times the exact result, which is labelled by "standard noise model". We observe that the results from the hardware agree very well with the emulator data. In particular the measured value at n = 10 is incompatible with the standard noise model by around 3 standard deviations. If we used the hardware data at n = 1 to calibrate the effect of the noise instead of the emulator data, the incompatibility would be even larger. In Figure 9 and Figure 10 we show the results of implementing the above-described algorithm for dynamical correlations at infinite temperature (i.e. assuming equal likelihood for all quantum states, as a mathematical simplification), comparing the exact values, the noiseless simulations and the results of the Quantinuum hardware emulator, without doing any noise mitigation. This emulator takes into account in particular depolarizing noise, leakage error and systematic coherent errors. We observe that the noisy values agree very well with the noiseless, despite the circuits containing several hundreds of two-qubit gates, showing a noise-resilience. In the graphs of Figures 9 and 10, expectation values tr [Z1(0)Z1(t)] / 2L (Figure 9) and tr [X1(0)X1(t)] / 2L (Figure 10) are computed with the shuffling quantum channel with the technique described in the section headed Supplemental Material below, as a function of t, with G(t) = elHtGe~ltH where H = H\sing on L = 8 sites. Noiseless simulations are shown as diamonds, whereas computations on a noisy emulator H1-2E are shown using circles and the exact values are shown as a continuous black line. We use Trotter steps dt = 0.1 to implement the time evolution. There are 1000 shots per circuit. Canonical ensemble expectation values Finally, we show that using a shuffling quantum channel with multiple ancillas, one can compute expectation values at finite temperature. Although this application suffers from exponential complexity and consequent system size and / or runtime requirements, the application shows the versatility of our channel in extracting physical information about a system. This can be expressed more generally, as follows We saw previously that the output of the shuffling channel can be expressed with two coefficients A and B, with B = —......—...........—. This expression is reminiscent of a partition function. As a comparison, expectation values at finite inverse temperature read -----If the eigenvalues of H are close to 0, the term cos2(H) will be close to e~™ for some appropriate value of p. If we could implement higher powers cos2N( / / ) with N integer, we could make this term arbitrarily close to e^H up to proportionality factors. It turns out that these higher powers can be implemented by generalizing the shuffling channel to multiple ancillas. To be more specific, let us consider a Hamiltonian H on L qubits, an observable O, an ancilla qubit v and an inverse temperature / ?. We define H' = arctan 0 + —........ 26N (27) on the system of L qubits plus one ancilla qubit v, where Zv denotes the Pauli matrix Z applied on the ancilla v, and where we introduced parameters e>0 and 0 >0, as well as an integer N. The additional ancilla v is notthe ancilla of the shuffling channel. When applying the shuffling channel as described below, this ancilla v will be considered part of the system. The parameters are chosen so that cos2N(H') converges to a Boltzmann weight. Namely, for TV larger than the norm of / 3H / 9, we have

[50] : cos2N(H') g-pH+eZvO (l+fl2)w ' (28) Now, as shown in Figure 6 and described in the section 'Generalization of the channel to multiple ancillas' described below, the shuffling quantum channel can be generalized into a channel Shuffle( / / , TV) that uses N ancillas as follows: one applies step 2 of the method implemented by the channel to each of the N ancillas, and then in step 3 one applies M1 ...ML on the L qubits only if a / / the N ancillas have been measured to be in state 0. An identical relation to (2) follows then with precise coefficients A and B given in the section 'Generalization of the channel to multiple ancillas'. In the particular case of H' in (27), because Zv commutes with H1, we have tr [p'Zv] = Atr [pZv] + B , with a _ 1 _ Tr | cos 2 A' ( / / ') | _ Tr |Z„Cos2W( / / ')| 2I+i ' D 2i+1 (29) (30) where Tr denotes the trace on the L qubits and the ancilla v. Hence, by repeating this generalized channel we obtain lim Tr [Zvpn] n—>oo Tr [Z„cos2W(H')] Tr [cos2W(H')] ’ (31) This prepares the ancilla in an ensemble that is close to a canonical ensemble, according to (28). Hence, performing the trace over the ancilla qubit v we obtain in this regime (32) where we now use tr to denote the trace over the L qubits, without the qubit v . Interestingly, the algorithm automatically computes the ratio of traces tr and tr \e^H} / ZL that can each of them be exponentially small. Using (28), (29) and (30), the number of rounds n required to reach this limit is l / log(l — tr [e @H] 2L(1+92)N . The time complexity is thus exponential in the system size L, which is the expected scaling for a finite-temperature algorithm that applies to generic Hamiltonians. However, this application shows the versatility of our shuffling channel to prepare qubits in a specific state. Discussion This patent specification provides evidence for the usefulness of mid-circuit measurements and classical conditioning on the measurement outcomes (also called feed-forward) as integral elements of quantum algorithms, and not only as quantum error correction subroutines. We introduced a "shuffling" quantum channel based on mid-circuit measurements and feed-forward with two verified properties. Firstly, it can be used to compute dynamical correlations at infinite temperature and finite temperature expectation values for any Hamiltonian. Although the algorithm for finite temperature expectation values is exponential in system size runtime, the application to dynamical correlations at infinite temperature is of current practical use. Secondly, it displays a noise-limiting behaviour, in the sense that it outputs a non-zero signal on a noisy hardware even when repeated an arbitrary number of times. We give analytical, numerical and experimental evidence on the Hl-1 trapped-ion quantum computer from Quantinuum for this noise-limiting property. This shuffling quantum channel shows that algorithms based on midcircuit measurements and feed-forward can support higher levels of noise than purely unitary circuits, making them particularly interesting for present-day quantum computers. In this specification, we have provided a first example of how non-error-corrected algorithms can avoid total depolarisation in the infinite-depth limit. Furthermore, although the algorithms presented here for finite temperature expectation values have an exponential time complexity (and this is unavoidable without further assumption on H), the inventors have noted the potential for improvements for local Hamiltonians H. Generalization of the channel to multiple ancillas In this Section we generalize the shuffling quantum channel to multiple ancillas. Namely, for an integer N we define Shuffle(H,W) through the following steps. 1. Pick a Pauli matrix at each site, i.e. ... ,ML 6 (I, X, Y, Z] uniformly at random. 2. Prepare N ancillas in state |0)ai... |0)ajv. For = 1,..., N, apply elHMi -MLXat. 3. Measure the N ancillas. Only if the measurement outcome is 0 for all the N ancillas, apply on the L qubits. The analysis of Shuffle(H) is straightforwardly generalized to the case N >1. For any Pauli string operator S = M'^.. M'L with M'i G {I, X, Y, Z}, we have Tr [Sp'] = ATr [Sp] + B , (33) with (34) x Scosp(BT)smN~p(Hy], and B = Tr [Scos2W(H)] Moreover, the probability p of measuring 0 in all the N ancillas is p = .....(36) When S commutes with H, we obtain ^ = 1_Tr[£^Wi {37) As noted above, the system on which we apply this quantum channel contains L qubitsand one extra ancilla v. It is thus a system with L + l qubits. Moreover the Hamiltonian H1 commutes with the particular choice of Pauli string S = Zv that we consider. This yields thus formula (30) above. Alternative wav of extracting the coefficients A and B In this Section we present an alternative way of computing coefficients A and B appearing in the shuffling quantum channel output in Equation (2) above. By applying twice Shuffle(H) on a density matrix p, and denoting p" the output density matrix, we have for any Pauli string S Tr fp"S] = A2Tr [pS] + AB + B . (38) It follows thus that we can extract A from the ratio A _ Tr[p"S]-Tr [p'S] Tr [p'S]-Tr [pS] ’ ' ' where we recall that p' denotes the density matrix after one single run of Shuffle(H). Then, B is given by B = Tr [p'S] - ATr [pS] . (40) Applying this process to the Hamiltonians Ht and Ht + p one can then sum the resulting .4's and obtain the dynamical correlation Cs(t). This way of computing Cs(t) presents sometimes a lower variance compared to the one quoted in the main text, although more involved. Let us indeed compute the variance A in the 1—Tr Fi / Sl2 estimation of A. 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Claims

1. A computer-implemented computation method comprising:generating a quantum channel, including operations to be performed with respect to quantum states, to represent the evolution of a physical system, wherein the quantum channel comprises quantum circuits adapted to perform at least a first computation step on a set of qubits including at least one ancilla qubit, followed by at least one mid-circuit state measurement performed on the at least one ancilla qubit, and to perform at least a second computation step that is conditional on a result of the mid-circuit state measurement, to generate a mixed state output of the quantum channel; andexecuting the at least one quantum channel on a set of qubits of a quantum computing apparatus, including performing at least a first computation step followed by performing the at least one mid-circuit measurement on an output of at least one ancilla qubit of the set of qubits, and performing the at least one conditional computation step on a subset of the qubits other than the measured ancilla qubit, thereby to generate a mixed state output containing information relating to the physical system.

2. A method according to claim 1, wherein the computer system comprises a classical computing apparatus and a quantum computing apparatus, with the generation of the quantum channel performed on the classical computing apparatus and the execution of the quantum channel performed on the quantum computing apparatus.

3. A method according to claim 1 or claim 2, wherein the at least one ancilla qubit is a single ancilla qubit.

4. A method according to claim 1 or claim 2, wherein the at least one ancilla qubit comprises a plurality of ancilla qubits.

5. A method according to any preceding claim, wherein the performance of at least a first computation step establishes a correlation between the at least one ancilla qubit and other qubits of the set of qubits, such that the result of the mid-circuit measurement performed on the at least one ancilla qubit is dependent on the first computation on the other qubits.

6. A method according to any preceding claim, wherein execution of the quantum channel is repeated, using the mixed state output of an execution of the quantum channel as an input to a next execution.

7. A method accordingto claim 6, wherein execution of the quantum channel is repeated multipletimes until the output mixed state has converged to a fixed point of the quantum channel.

8. A method according to claim 6, wherein the qubits of the set of qubits are prepared in states defined in probability density matrix representing possible states of the physical system prior to performing the first computation step, and execution of the quantum channel outputs a modified probability density matrix representing possible states of physical system, wherein the output matrix converges to an equilibrium state of the quantum channel after repeated execution of the quantum channel.

9. A method according to any preceding claim, wherein the generation of a quantum channel is performed in response to an input representation of a physical system for which the computation method is required.

10. A method according to claim 9, wherein the representation of the physical system is a Hamiltonian.

11. A method according to any preceding claim, wherein the quantum channel is adapted to calculate expectation values or dynamical correlations for operators and / or quantum states of a physical quantum system.

12. A method according to any preceding claim, wherein the representation of a physical system is an Hamiltonian or other mathematical representation of a quantum physical system and the quantum channel is adapted to calculate expectation values or dynamical correlations for operators and / or quantum states of a quantum physical system.

13. A quantum computing system comprising:a first computing apparatus configured to generate a quantum channel for performing a sequence of computational steps for computing information relating to a physical system, the quantum channel comprising quantum circuits adapted to perform at least a first computation on a set of qubits including at least one ancilla qubit, and wherein the quantum channel comprises at least one mid-circuit state measurement performed on the at least one ancilla, and the quantum channel comprises at least one computation step that is conditional on a result of the mid-circuit state measurement, and wherein the quantum channel is adapted to generate a mixed state output; anda quantum computing apparatus comprising a set of qubits for executing the quantum channel, including performing at least a first computation step on the set of qubits including at least one ancilla qubit, followed by performing the at least one mid-circuit measurement on an output of at least one ancilla qubit of the set of qubits, and performing the at least oneconditional computation step on a subset of the qubits other than the measured ancilla qubit, thereby to generate a mixed state output containing information relating to the physical system14. A system according to claim 13, wherein the quantum computing apparatus includes L 5 computational qubits and one or more ancilla qubits, and the quantum channel is configured to perform at least a first operation on a first set of qubits including the ancilla qubit(s), and to perform one or more mid-circuit measurements on the ancilla qubit(s), and to perform at least one conditional computation step on the L qubits excluding the measured ancilla qubit(s).10 15. A system according to claim 14, wherein a Pauli matrix is allocated to each of the Lcomputational qubits, and the one or more ancilla qubits are initialized in a first state, and then a first operation is performed on the L qubits and the one or more ancilla qubits based on the allocated Pauli matrix, and wherein a mid-circuit measurement is performed on the one or more ancilla qubits, and wherein the respective Pauli matrix is applied to each of the L computational15 qubits on condition of a specific measurement result of the one or more anciIlas qubits.

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