Quantum computing system and method for noise mitigation

Abstract

Described are computational methods and systems in which 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 resilience to noise and decoherence, and maintains quantum states with useful properties even for infinitely deep noisy circuits. The system and computation methods can be used, for example, to compute dynamical correlations at infinite temperature and canonical ensemble expectation values for any Hamiltonian.

Description

[0001] A^orney ref: 65193 PCTQuantum Compu^ng System and Method for noise mi^ga^onTechnical Field The present disclosure relates to noise mi^ga^on within a quantum compu^ng system, such asa hybrid compu^ng system including a combina^on of a classical binary computer coupled to aquantum computer. Moreover, the present disclosure relates to a method for performingcomputa^ons within a quantum compu^ng system. Furthermore, the present disclosure relatesto so^ware products recorded on machine-readable media, wherein the so^ware products include program code which is executable to control genera^on of quantum circuits and opera^on of a quantum compu^ng system. Background The main obstacle to large-scale quantum compu^ng is the imperfect realiza^on of quantumopera^ons [1, 2] and a longstanding problem in the development of quantum computers hasbeen how to maintain quantum performance, such as quantum coherence, in the presence ofnoise. Various sources of noise, such as heat or electromagne^c radia^on, can cause unwanted disturbances to the devices carrying out quantum computa^ons, effec^ng control of qubits andinterfering with reliable and accurate execu^on of quantum opera^ons as well as causinginforma^on within the signal output to be lost. Quantum computers have recently become available as noisy intermediate-scale quantum (NISQ) devices, and quantum computers have alsobeen 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 limita^ons in the types of algorithms that are executable on currently-available NISQ devices,where considera^ons such as the circuit depth or the available resources (e.g., limited numberof qubits and a lack of qubitfidelity) need to be taken into account. Therefore, mi^ga^on of theproblems of noise is a significant technical problem associated with NISQ devices. Various methods have been proposed in published scien^fic literature that account for errorsthat arise when performing computa^on on NISQ devices, such as error detec^on and errorcorrec^on methods, alongside efforts to improve qubitfidelity. However, due to limita^ons inthe scalability and accuracy of current NISQ devices, exis^ng quantum computers can onlyimplement quantum error correc^on protocols on few logical qubits with limited circuit depth.In the presence of noise, the signal output by a quantum circuit is generally exponen^allya^enuated with the number of gates [3-7], causing exponen^ally increasing overheads which haslimited the prospects for u^lity of currently available quantum computers.A^orney ref: 65193 PCTSummaryThe present disclosure provides improved methods and systems for performing noise resilientcomputa^ons using a quantum compu^ng system. Described below are quantum compu^ngsystems and computa^on methods in which a quantum channel describing the evolu^on of aphysical system incorporates one or more mid-circuit measurements with at least onesubsequent computa^on step which is condi^onal on the measurement outcome, wherein themid-circuit measurements and condi^onal computa^on steps are integral computa^onalprimi^ves of a quantum channel that outputs a mixed state. The result of this integra^on is thatthe quantum circuit that prepares the mixed state is naturally noise-resilient and avoids completedecoherence, differing from conven^onal approaches to error mi^ga^on. The quantumcompu^ng systems and computa^on methods can provide a non-zero signal output that containsuseful informa^on about the evolu^on of a physical system, even for a large circuit depth or a^eran arbitrarily large number of repeats in the presence of noise.Afirst computer-implemented computa^on method comprises: genera^ng a quantum channel, including opera^ons to be performed with respect toquantum states, to represent the evolu^on of a physical system, wherein the quantum channelcomprises quantum circuits adapted to perform at least afirst computa^on step on a set of qubits(or qudits) including at least one ancilla qubit (or qudit), followed by at least one mid-circuit statemeasurement performed on the at least one ancilla qubit (or qudit), and to perform at least asecond computa^on step that is condi^onal on a result of the mid-circuit state measurement, togenerate a mixed state output of the quantum channel; andexecu^ng the at least one quantum channel on a set of qubits or qudits of a quantum compu^ng apparatus, including performing at least afirst computa^on step followed byperforming 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 condi^onal computa^onstep on a subset of the qubits (or qudits) other than the measured ancilla qubit (or qudit), therebyto generate a mixed state output containing informa^on rela^ng to the physical system. Preferably, the mixed state output is input to a subsequent execu^on of the quantum channel,and execu^on of the quantum channel is repeated using the mixed state output of the previousitera^on. Quantum channels are used to describe the evolu^on of physical or computa^onal quantum systems as they interact with their surroundings or undergo various opera^ons. Quantum channels represent the transforma^on of quantum states due to those interac^ons (with theenvironment or inten^onal opera^ons within a quantum system), for example using probabilitydensity matrices to describe the evolu^on of the quantum channel.A^orney ref: 65193 PCTThe quantum channel execu^on may be repeated mul^ple ^mes to stabilize a mixed state outputon the qubits of the quantum compu^ng system. The method would output a mixed state(obtained as a ‘fixed point’ or equilibrium state of the quantum channel) even in the absence ofnoise, but is especially useful for mi^ga^ng the effects of noise as thefixed point varies onlysmoothly with noise strength such that the mixed state that is produced is not a completely depolarised state. Herea^er in this patent specifica^on, any reference to qubits should be interpreted as a reference to qubits or qudits, as the inven^on is equally applicable to both. At least one mid-circuit measurement and at least one computa^on step that is condi^onal onthe measurement result are preferably implemented as integral computa^onal primi^ves withinthe quantum channel itself. This integra^on builds noise-resilience into quantum computa^ons.The output of execu^on of the at least one quantum channel on a quantum computer is a mixed state output, which can provide informa^on rela^ng to the physical system even in the presenceof noise, par^al decoherence, or imperfect measurements. A mixed state is a sta^s^cal ensembleof quantum states, which represents the probabili^es of finding the system in different purestates, for example as described by probability density matrices. Both pure and mixed states canbe described mathema^cally using the density matrix representa^on. Methods according to the inven^on can generate a mixed state output of the quantum channel on the system qubits thatcannot be produced by non-unitary opera^ons (without incurring a run^me that is exponen^allylarge in the number of qubits) and improves upon purely unitary uncondi^onal opera^ons, butthe opera^ons that are condi^onally performed on the system qubits based on the ancilla qubitmeasurement are preferably implemented as unitary opera^ons. Condi^onal performance ofnon-destruc^ve unitary opera^ons preserves the probability of different outcomes.The computer system may comprise a classical (digital) compu^ng apparatus and a quantum compu^ng apparatus, with the genera^on of the quantum channel performed on the classical compu^ng apparatus and the execu^on of the quantum channel performed on the quantum compu^ng apparatus. A method of computa^on preferably comprises providing a representa^on of a physical system as an input to a computer system, for use in genera^on of a quantum channel. The representa^onof the physical system includes at least one operator to perform opera^ons with respect to aquantum state of the represented physical quantum system. This representa^on of the physical system may be a Hamiltonian, for example for mathema^cally represen^ng the energy states and interac^ons of a physical quantum system, or may be a mathema^cal representa^on of theenergy states of a classical system. A Hamiltonian can be used to generate one or more quantumA^orney ref: 65193 PCTcircuits for ac^ng on a set of qubits of the quantum computer system. Ini^alisa^on of a quantumstate followed by execu^on of the quantum circuits on the qubits and measurement of qubit outputs provides informa^on about the physical system represented by the Hamiltonian.A quantum channel with integral mid-circuit measurements, and integral condi^onal execu^onof one or more computa^onal steps within the quantum channel, differs from known quantumerror mi^ga^on techniques that rely on separate error-checking subrou^nes that do not performany computa^on beyond their error checking. It is known to provide subrou^nes for mid-circuit measurement that are used to detect errors and to decide whether to restart the en^recomputa^on from the beginning (i.e. if certain errors are a stopping condi^on), where thefinaloutput a^er postselec^on is a pure state. Quantum-error-corrected circuits would output a purestate in the absence of noise. However, the present inven^on differs by its integra^on of mid-circuit measurements performed on one or more ancillas with condi^onal performance of thesubsequent computa^on also implemented as an integral part of the quantum channel itself, andby outpu^ng a mixed state. The condi^onal computa^on steps can be implemented ascomputa^onal primi^ves within the quantum channel, in contrast to separate error checkingsubrou^nes. The result is an integral noise-resilience that avoids some of the limita^ons whicharise when implemen^ng dedicated quantum error detec^on / correc^on subrou^nes on NISQ devices.A method according to the present inven^on provides a noise-resilient way of preparingquantum states, as the output from the described quantum circuits, in the form of a mixed state.This state prepara^on could be used for a wide range of subsequent computa^ons, performed on the same or different qubits. One of the poten^al applica^ons of the present inven^on is for ini^alisa^on of qubits to a specific mixed quantum state, in advance of a subsequent quantum compu^ng opera^on. However, the present inven^on’s combina^on of mid-circuit measurement and “feed forward” performance of opera^ons condi^onal on this measurement may be integrated within a wide range of computa^ons, where it is used to add noise resilience to that computa^on. The prepara^on 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 correc^on rou^nes which only delay the depolarizing). Contrary to previously known codes, implementa^ons of the present inven^on can output a non-zero signal containing useful informa^on about the transforma^on / evolu^on of a physical system, even a^er anarbitrarily large number of repeats in the presence of noise.For some implementa^ons, the inven^on provides this noise resilient behaviour with a low qubitoverhead, which is advantageous when execu^ng on NISQ devices. For example, the inven^on isA^orney ref: 65193 PCTimplementable with an overhead of only one ancilla qubit, or alterna^vely using mul^ple ancillas,in combina^on with the qubits that are used to perform the complete computa^on. This lowqubit overhead is possible when protec^ng a specific outcome within the quantum channel, contrary to known error-correc^on rou^nes which are intended to be applicable to a variety of different circuits. Thefixed point of the channel encodes informa^on about the physical system and varies smoothly with noise, making the architecture of the quantum computer an important considera^on to implement useful algorithms in the NISQ era. Ion-trap based quantumcomputers have compe^^ve advantages for the implementa^on of algorithms based on mid-circuit measurement, and feed-forward. Ion-trap based quantum computers achieve some of thehighest gatefideli^es, lowest SPAM (state prepara^on and measurement) errors and longestcoherence ^mes in the current era. Ion-trap based quantum computers have achieved highquantum volume metrics and allow for itera^ve algorithms to be implemented with the smallestpossible error propaga^on rate in the absence of error correc^on techniques. Addi^onally, ion- trap based quantum computers developed by Quan^nuumTMhave large qubit connec^vity, evenup to all-to-all qubit connec^vity. This is advantageous for circuits containing mid-circuitmeasurement and feed-forward as it greatly simplifies the task of mapping algorithms to physical qubits, such as by reducing the number of swap opera^ons required to account for the dynamicnature of adap^ve quantum circuits based on the result of mid-circuit measurements andreducing the impact of mid-circuit measurements on the computa^on (mid-circuit measurements can affect other qubits in a superconduc^ng quantum computer that requiresinterac^on with nearest neighbour qubits). Therefore, ion-trap based quantum computers arean architecture well suited to the use of the naturally noise-resilient algorithms described in thisspecifica^on.The noise resilience of the disclosed algorithms have been demonstrated on the Quan^nuumTMH1-1 computer, where expecta^on value outcomes were found to agree well with noiselesscircuits without the use of error mi^ga^on techniques. In an embodiment, execu^on of the quantum channel comprises the following steps. Firstly, a classical random variable is generated and the qubits are ini^ally prepared in a state independentof any previous calcula^ons or measurement outcomes. An operator that depends on therandom variable is applied to a set of qubits of a quantum compu^ng apparatus including one or more ancilla qubits. This operator can be a unitary operator that entangles the one or more ancilla qubits with the other qubits of the set, or another opera^on that couples (i.e. establishesa correla^on between) the ancillas and the other qubits. For example, this coupling may use a^me evolu^on operator e^{i H X} applied to a set of qubits including the one or more ancillas,A^orney ref: 65193 PCTwhere 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 ormore ancillas, with the results of the measurement used for condi^onal control of one or moresubsequent computa^on steps performed on a subset of the qubits excluding the measuredancilla(s). For example, an opera^on may be performed on L qubits and one ancilla qubit, where L is a number of qubits used for a computa^on; a mid-circuit measurement may be performed on the one ancilla qubit; and then the computa^on may be con^nued on the L qubits condi^onalon the result of the mid-circuit measurement. The mixed state output of thefirst itera^on canthen be used as an ini^al state for subsequent itera^ons of the execu^on of the quantumchannel, repea^ng un^l a rela^vely stable mixed state output is achieved or repea^ng anarbitrarily large number of ^mes.The computa^on method may be applied, with a low qubit overhead, in the computa^on ofdynamical correla^ons with an assump^on that all outcomes are equally likely (referred to asdynamical correla^ons at “infinite temperature”). The described computa^on methods are alsoapplicable for performing computa^on of finite temperature expecta^on values for a Hamiltonian, if we use mul^ple ancillas, and can be used for studying the proper^es of condensedma^er systems and NMR systems such as the effect of temperature. For example, Gibbs statesrepresen^ng the equilibrium state of a quantum system at a given temperature are mixed states and the computa^onal methods described herein can be used with Gibbs states to inves^gate phase transi^ons such as whether there is a superconduc^ng phase of a system atfinite temperatures. Afirst quantum compu^ng system comprises: afirst compu^ng apparatus configured to generate a quantum channel for performing asequence of computa^onal steps for compu^ng informa^on rela^ng to a physical system, thequantum channel comprising quantum circuits adapted to perform at least afirst computa^onon a set of qubits including at least one ancilla qubit, and wherein the quantum channelcomprises at least one mid-circuit state measurement performed on the at least one ancilla, and the quantum channel comprises at least one computa^on step that is condi^onal 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 compu^ng apparatus comprising a set of qubits for execu^ng the quantumchannel, including performing at least afirst computa^on step followed by performing the atleast one mid-circuit measurement on an output of at least one ancilla qubit of the set of qubits,and performing the at least one condi^onal computa^on step on a subset of the qubits otherthan the measured ancilla qubit, thereby to generate a mixed state output containinginforma^on rela^ng to the physical system. Preferably, the mixed state output is input to aA^orney ref: 65193 PCTsubsequent execu^on of the quantum channel, and execu^on of the quantum channel is repeated using the mixed state output of the previous itera^on. The quantum channel may be implemented with only a single ancilla qubit, or with mul^ple qubits. Thefirst compu^ng apparatus may be a classical digital compu^ng apparatus. Brief Descrip^on of Figures Described below are methods, systems, apparatus and computer programs for implemen^ng the present inven^on, provided by way of example only, with reference to the accompanying drawings in which: Figure 1 is a schema^c representa^on of an example quantum compu^ng system, including aclassical compu^ng apparatus coupled to a quantum compu^ng apparatus and arranged tocooperate to execute computa^onal tasks, involving the genera^on and execu^on of a quantumchannel;Figure 2 shows a sequence of steps of an example method implemen^ng the inven^on;Figure 3 represents a quantum channel being executed on a set of qubits including: pre-measurement steps, followed by mid-circuit measurement of an ancilla qubit, followed by feed- forward condi^onal performance of computa^ons on system qubits other than the ancilla qubit;Figure 4 shows steps of a method implemen^ng some of the steps of Figure 2;Figure 5 represents the performance of opera^ons of a quantum channel using a single ancillaqubit;Figure 6 represents repeated performance of opera^ons of a quantum channel using a singleancilla;Figure 7 represents the performance of opera^ons of a quantum channel using mul^ple ancillaqubits;Figure 8 is a graphical representa^on of the noise-mi^ga^on effects of a quantum channelaccording to the inven^on; andFigures 9 and 10 are graphical repreenta^ons of dynamical correla^on results for exampleA^orney ref: 65193 PCTquantum channels according to the inven^on. Detailed Descrip^on 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 integralcomputational primitives within a quantum channel – i.e. within a quantum computing circuit thatcomputes 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 isconditional on the measurement outcomes are implemented as integral computational tools(rather than error correcting subroutines), making the computations naturally noise-resilient inthe sense that, even in the limit of an infinitely deep circuit, there remains a non-zero signal thatcontains information on the task fulfilled by the algorithm.Figure 1 is a schema^c illustra^on of a hybrid quantum compu^ng system 10 including a classicalcompu^ng apparatus 20 coupled to a quantum compu^ng apparatus 30. The classical compu^ngapparatus and the quantum compu^ng apparatus are configured to operate in tandem to execute computa^onal tasks involving the genera^on and execu^on of a noise resilient quantum channel. Specific quantum channels for implemen^ng the present inven^on are described below.As shown in Figure 2 and Figure 3, for a single execu^on of the channel, the steps of a methodfor implemen^ng the inven^on can be represented as a simple sequence of steps. The methodstarts with receipt 100 of data such as an input Hamiltonian or other informa^on about aquantum system for which a computa^on is required. A quantum channel is generated 110 by aquantum circuit builder 22 running on the classical compu^ng apparatus 20 of the hybrid quantum-classical system 10. This quantum channel comprises quantum circuits for performingat least afirst computa^on on a set of qubits 32,34 of the quantum compu^ng apparatus 30,which we will refer to as a number L of system qubits 32 and an ancilla (or auxiliary) qubit 34. Thequantum channel is generated to include, as computa^onal primi^ves integral to the quantumchannel: a mid-circuit measurement 132 carried out on the ancilla qubit 34; and at least onesubsequent computa^on 142 to be performed on the L system qubits which is condi^onal on theoutcome of the ancilla measurement. Following genera^on 110 of the quantum channel, thegenerated quantum circuits are executed on the qubits 32,34 of the quantum compu^ngapparatus 30, including performing 120,126 at least afirst computa^on on the L qubits and ancillaqubit, and then performing 130,132 a mid-circuit measurement on the ancilla qubit. Addi^onally,the subsequent computa^on is performed 140,142 on the L qubits, condi^onal on the outcomeof the measurement result of the ancilla qubit. The result of the computa^on is prepared as amixed state output. This sequence of steps can then be repeated, using the mixed state outputof thefirst itera^on as an input for the next itera^on.Some steps of the method are shown in more detail in Figure 4, by way of example. Steps 110 and120 of Figure 2 can be implemented using steps 122, 124 and 126 of Figure 4; and step 130 ofA^orney ref: 65193 PCTFigure 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 ancilla qubit in a first initialized state, |0〉^, 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 moresubsequent computations 142 of the quantum channel are made conditional on the determinedresult 134 of that measurement – i.e. only if the measurement outcome is a specific result such as0, 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 aquantum channel as defined below and shown schema^cally in Figure 3. We start with an ini^aldensity matrix 200 of the L qubits, and a single ancilla qubit 210, as shown in Figure 5. Execu^onof the quantum channel 300 outputs a new density matrix 200’ and one observable ancilla qubitstate 210’. Figure 6 is similar to Figure 5, but with the quantum channel repeated n ^mes. Figure7 is also similar to Figure 5, but with a plurality of ancilla qubits. These op^ons are describedbelow.The inven^on has wide applicability but, as afirst 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 ^mes, even in the presence of noise. Crucially, this non-zero signal also carries non-trivial, physically interes^ng informa^on: We show that this quantum channel can be used to computefinite temperature expecta^on values for anyHamiltonian ^ , as well as dynamical correla^ons at “infinite temperature” t^ [^^ ^^^^^^^^^]where ^ is any Pauli string. The condition of “infinite temperature” is a conceptual condition inwhich all states are assumed to be equally likely. The inventors have also shown that the describedmethod can be implemented to prepare certain highly quantum correlated density matrices in anoise-resilient way. The use of mid-circuit measurements to perform non-unitary opera^ons is known, for example as a means of post-selec^ng par^cular measurement outcomes [14-23], to prepare specific, analy^cally tractable quantum states more efficiently [24-29], or on top of another algorithm to mi^gate 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 par^ally reset the state [33-38]. The present inven^on differs from these known techniques by the use of mid-circuitmeasurement and feed-forward as computa^onal primi^ves to obtain non-trivial physicalproper^es which endow the computa^on with a natural noise-limi^ng property. This noise-limi^ng behaviour has been demonstrated analy^cally with specific noise models, numericallywith noisy simula^ons, and experimentally on ion-trap quantum computers from Quan^nuumTM.The shuffling quantum channel Afirst quantum channel is described below by way of example, with reference to Figures 3 andA^orney ref: 65193 PCT4. The effect of the channel on expecta^on values is given by Equa^on (2) below. The inventorshave determined that a quantum channel which implements mid-circuit measurements withfeed forward condi^onality within the channel can be used to compute dynamical correla^onsat infinite temperature. An unusual property of this channel is that when repeated several ^mesin presence of noise, it s^ll outputs meaningful non-zero signal.For a generic Hamiltonian ^ on ^ qubits, we define the quantum channel Shuffle(^) by thethree following steps. 1. Pick a Pauli matrix at each site, i.e. ^^, ... , ^^ ∈ {^, ^, ^, ^} uniformly at random.2. Prepare an ancilla in state On the ^ qubits and the ancilla, apply (1)3. Measure the ancilla. Only if the measurement outcome is 0, apply ^^... ^^ onthe ^ qubits.Let us denote ^ the ini^al density matrix of the ^ qubits, and ^′ the new density matrix a^erone run of Shuffle(^). We are going to show that for any Pauli string operator ^ = ^′^... ^′^with ^′^ ∈ {^, ^, ^, ^}, we have:t^ [^^′] = ^t^ [^^] + ^ , (2)with ^^ [^^^^(^)^^^^(^) ^^ = ] ^^ [^^^^ (^)]^^, ^ =^^. (3)Moreover, the probability ^ of measuring 0 in the ancilla is= ^^ [^^ ^^ ^ (^)]^^. (4)ProofFrom step 1 of Shuffle(H) above, wefix some Pauli matrices at each site , ^^ ∈ {^, ^, ^, ^}.We will denote ^ = ^ ... ^ and ℳ the set ^^ ^ of all the 4 Pauli strings. The operator ^ =^^^^^^is diagonal in the eigenbasis of ^^, with matrix elements ^^that are operators on the ancilla and that read with ^^the corresponding eigenvalue of ^^. By preparing the ancilla in the state |0〉, applying^ and measuring the ancilla, we act on the ^ qubits with the operator 〈0|^|0〉 = cos(^^) ifwe measure 0 , and with = ^sin(^^) if we measure 1. It follows that the densitymatrix ^′^ that we obtain a^er the steps (2) and (3), assuming have been pickedat step (1), is ^′^ = sin(^^)^sin(^^) + ^cos(^^)^cos(^^)^= sin(^ )^ ^sin(^^) + cos(^)^^^cos(^) .A^orney ref: 65193 PCTTaking into account all the 4^ possibili^es of at step 1 of Shuffle(H), the densitymatrix ^′ a^er one run of Shuffle(^) is the sum of all the ^′^’s divided by 4^. We now note that on the second term of ^′^, this sum precisely implements a completely depolarizingchannel

[0013] . Namely, for any density matrix ^ we have It follows ^′ = ^sin We note that the feed-forward in step 3 of Shuffle(H) is important to obtain a non-trivial densitymatrix, following the mid-circuit measurement. Had we always applied ^ on the qubitsindependently of the measurement outcome of step 2 of Shuffle(H), the density matrix ^′ wouldhave been the completely depolarized state 1 / 2^.Let us now consider a given Pauli string operator ^′ ∈ ℳ, and denote the expecta^on values^ = t^ [^′^] , ^′ = t^ [^′^′] . (9)Mul^plying (8) by ^′ and taking the trace, we make appear in the sum the termst^ [^′sin(^^)^sin(^^)]. Decomposing ^ in the basis of Pauli strings we write ^sin(^^)][^^′^sin(^)^^(11) ′′^sin(^)] .Let us performfirst the sum over ∈ {^, ^, ^, ^} appearing in (8). We note that ^^^^^^^either ^^is ^(if = ^, or ^ ^^ = ^, or ^^ = ^^^), or −^^^(in the remaining cases). If ^^^≠ ^^^^, then at least one of the two is not ^, say ^^^ ≠ ^. If ^ ^^^ = ^, then one sees that one getsan amplitude t^ [^′sin(^)^′′sin(^)] with a + sign when a − sign in the two remaining cases. Hence this term vanishes once summed over If ^^^ and ^′′ ≠ ^, the same reason ^^^ ing applies. If both ^′^ , ^^ ≠ ^, one sees that one gets a +sign when ^^ = ^ and ^^ ≠ ^ ^^ , ^ ^^^ , and a − sign in the two remaining cases. Hence thisterm also vanishes once summed over It follows that once we sum over all the ^^, onl terms where ^^y ^= ^^^^give a non-zero contribu^on. This yields No^ng that we have ^ = ^^^ 2^, we obtain thusA^orney ref: 65193 PCTwhich is precisely (2). To compute the probability ^ of measuring 0 in the ancilla, we note thatif has been picked at step 1 of Shuffle(H), then this probability ^^ is^^ = t^ [^cos^(^^)] . (14)Summing over ^^, ... , ^^, this yields ^^^^(^) Using in this equa^on the rela^on (7) for the density matrix^^ [^^^^(^)]we obtain (4). Dynamical correla^ons at infinite temperature Let us show how the shuffling quantum channel can be used to compute dynamical correla^onsof Pauli strings ^ = ^^... ^^ at infinite temperature, defined as These quan^^es are relevant for transport proper^es and have a^racted par^cular interest recently [39-48]. To compute this quan^ty with the shuffling quantum channel, we write Let us consider ^ an arbitrary ini^al density matrix for which we know t^ [^^]. We then applyon ^ either Shuffle(^^) with probability 1 / 2 , or Shuffle(^^ with probability 1 / 2. Wemeasure the Pauli string ^ and denote t^ [^′^] the average outcome. Using formula (2), wehave thus with ^^, ^^ the coefficients (3) corresponding to the Hamiltonian ^^ + ^. From (3) we thusfind Hence, one can directly compute the value of ^^(^) provided we chose ^ such thatt^ [^^] ≠ 0. There are many other similar ways of extrac^ng ^^(^) from this quantum channel

[0049] . This algorithm for dynamical correla^ons at infinite temperature presents advantages compared to previously exis^ng techniques. Compu^ng the trace over the Hilbert space through a purifica^on requires to double the number of qubits, whereas here there is only one ancilla qubit overhead. Implemen^ng the trace through a Haar random state prepara^on

[0049] has a gate overhead to prepare the Haar random state, and also a shot overhead because it computes the square of the dynamical correla^ons and can only treat one site at a ^me, which increasesA^orney ref: 65193 PCTconsiderably the variance. Finally, for ^ single Pauli matrices, evalua^ng the trace as a sum overproduct states in the basis of ^ can have a large sta^s^cal sampling overhead if dynamical correla^ons are small in this basis, whereas our algorithm is basis-independent.Dynamical correla^ons at different la^ce sites such as t^ [^ℓ^^(^)^^(0)] / 2^can be obtained as well with the following modifica^on of the channel. Just a^er applying in thesecond step of the channel, we apply the operator that swaps qubits 1 and ℓ + 1 condi^onedto ^^being −1. Density matrix prepara^on The density matrix constructed previously in (19) contains informa^on about the physics of thesystem through ^^(^) , but also on the ini^al density matrix through t^ [^^] . This can beproblema^c for strings ^ involving different Pauli matrices at different sites, since theprepara^on of ^ such that t^ [^^] ≠ 0 for mul^ple such ^ would become non-trivial.The shuffling quantum channel through the rela^on (2) actually enables one to prepare density matrices with high quantum correla^ons independently of the ini^al density matrix. Withreference tofigures 3 and 5, let us repeat ^ ^mes the channel Shuffle(^) on an ini^al densitymatrix ^^, deno^ng ^^ the resul^ng density matrix. Equa^on (2) yields a geometric series forany string ^ with ^, ^ given in (3). In par^cular, since |^| < 1 (except for very specific choices of ^, ^where we would have |^| = 1) when ^ → ∞ we obtain^i→m^t^ [^^^ l^] =^^^, (21)which does not depend on the ini^al density matrix. Moreover the convergence is exponen^allyfast in ^ with rate log1 / |^|. Repeated execu^on of the Shuffle(H) channel provides a highlylong-range quantum correlated state whose expecta^on values contain physical informa^on about the system studied. Noise-limi^ng behaviour: analy^csAs noted above, the algorithms based on the repe^^on of the shuffling quantum channel are tosome extent resilient to noise. To give evidence for this claim, let usfirst consider a simple noisemodel where a global depolarizing channel is applied on the system qubits a^er each run of Shuffle(H). Namely, the density matrix ^^a^er applying Shuffle(H) is sent to Eq (2) thus becomesA^orney ref: 65193 PCTt^ [^^′^^^^^] = A′t^ [^^] + ^′ ,with ^^ = (1 − ^)^ and ^^ = (1 − ^)^ . It follows that a^er itera^ng the shuffling channelwith the noise round, we obtain the following expecta^on values within the density matrix ^ li→m^t^ 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 ancilla. Let us explain why this noise-limi^ng behaviour is directly related to the use of mid-circuit measurements and feed-forward. A circuit involving only unitary opera^ons and measurementswithout feed-forward can always be wri^en as the composi^on of density matrix maps ^(^) =^^^^ with ^ unitary or ^(^) = ^^^^^^with ∑^^^^^^ = Id . These maps bothsa^sfy ^(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^(^) = ∑^ ^^^^^^^^^^^with ^^unitary, for which the iden^ty is (generically) not afixed point. This thus prevents the completely depolarized state to be reached in the limit of large number of gates. Together with the previous calcula^on with the global depolarizing noisechannel, this yields the following intui^ve explana^on of the noise-resilience of our algorithm.The resul^ng density matrix a^er a large number of rounds converges to thefixed 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, incorpora^ng noise in the process (under any form) will only smoothly change the quantum channel, and so only smoothly change thefixed point, instead of sending it to the completely depolarized state. Thenoise-resilience of the prepared density matrix is thus built into the algorithm itself.The same calcula^on as above can be applied to slightly less simple noise channels. Let usconsider now a noise model where a depolarizing channel with amplitude ^ is applied to eachqubit a^er the applica^on of the operator ^^^^^^appearing in Shuffle(H). This noise model isequivalent to applying a ^, ^, ^ error each with probability ^ / 4 on the qubit, and applying noerror with probability 1 − 3^ / 4. We considerfirst the effect of applying ^ on qubit ^ a^erstep (2) of Shuffle(H). This is equivalent to applying ^ ^^^^ ^^ on ^^^^^ with ^^ = ^^^. Deno^ng^^′ the density matrix obtained a^er steps (1), (2) and (3) when an ^ error has occurred at site^, then in terms of the noiseless value ^′ in (8) one has exactly^^′ = ^^^′^^ . (23)We thus obtain t^ [^^^′] = t^ [^^^^^^′] for a Pauli string ^.A^orney ref: 65193 PCTSince ^^ either commutes or an^commutes with ^, we have t^ [^^^^^^′] = ±t^ [^^′]. The +occurs with probability 1 − ^ / 2 and the – sign with probability ^ / 2. Hence the total noisydensity matrix ^′^^^^^a^er one round sa^sfies, taking into account the different possible errors: Deno^ng ^^^^^^,^the density matrix obtained a^er ^ noisy applica^ons of Shuffle(H), the limi^ng value reached in the limit of an infinite number of applica^ons is ^ li→m^t^ which isfinite, even if an infinite number of gates has been applied. The same result as for theglobal depolarizing channel is recovered. Moreover in this case, the noise level ^ appearing hereonly refers to the probability of error per site. Noise-limi^ng behaviour: experiments and simula^ons We now present numerical and experimental evidence of the noise resilience of our algorithm.We consider the 1D Ising model in a transversefield ℎ = 1 and ^ = 0.1 on ^ = 8 sites withperiodic boundary condi^ons We first present evidence for the noise-resilience of the simple shuffling quantum channel ^ Shuffle(^) . We ini^alize the ^ qubits in |0〉 and measure ^ ∑^^^^^^^^^^a^er ^ rounds of Shuffle(^). We chose this observable ^^would have a zero expecta^onvalue at either small or large number of rounds. We show in Figure 8 the results of noiselesssimula^ons, noisy simula^ons with the H1 emulator from Quan^nuumTM, and actual hardwareimplementa^on with the H1-1 machine from Quan^nuum for number of rounds ^ = 1 (185two-qubit gates and 782 one-qubit gates) and ^ = 10 (1850 two-qubit gates and 7820one-qubit gates). Figure 8 is a graphical demonstra^on of the noise-limi^ng property of the shuffling quantumchannel. Expecta^on values ^^ ≡^ ^∑^^^^ are plo^ed as a func^on of the number ^of calls to the quantum channel referred to herein as Shuffle(H), with an example Hamiltonian^ as disclosed in (33), and using 5 Tro^er steps to implement ^^^. There are 1000 shots percircuit for simula^ons, and 500 shots per circuit for hardware. The errors bars indicate ± onestandard devia^on. A^er compila^on, there are 185^ two-qubit gates and 782^ one-qubitgates as a func^on of ^. It is known that noise on quantum computers can be well accounted for, in afirst approxima^on, by a^enua^ng the signal with a factor equal to the product of thefideli^es of the gates enteringthe circuit. We evaluate the effect of the noise by measuring the signal a^er one round ^ = 1A^orney ref: 65193 PCTon the emulator with a large number of shots, and comparing it to the exact value. From this, weobtain an a^enua^on per round 1 − ^. We plot then in Figure 8 the effect of the noise assumingit can be modelled by an a^enua^on factor (1 − ^)^ at round ^ ^mes the exact result, whichis labelled by “standard noise model". We observe that the results from the hardware agree very well with the emulator data. Inpar^cular the measured value at ^ = 10 is incompa^ble with the standard noise model byaround 3 standard devia^ons. If we used the hardware data at ^ = 1 to calibrate the effect ofthe noise instead of the emulator data, the incompa^bility would be even larger.In Figure 9 and Figure 10 we show the results of implemen^ng the above-described algorithm fordynamical correla^ons at infinite temperature (i.e. assuming equal likelihood for all quantumstates, as a mathema^cal simplifica^on), comparing the exact values, the noiseless simula^ons and the results of the Quan^nuum hardware emulator, without doing any noise mi^ga^on. This emulator takes into account in par^cular depolarizing noise, leakage error and systema^c 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, expecta^on values t^ [^^(0)^^(^)] / 2^(Figure 9) andt^ [^^(0)^^(^)] / 2^(Figure 10) are computed with the shuffling quantum channel with the technique described in the sec^on headed Supplemental Material below, as a func^on of ^, with^(^) = ^^^^^^^^^^ where ^ = ^^^^^^ on ^ = 8 sites. Noiseless simula^ons are shown asdiamonds, whereas computa^ons on a noisy emulator H1-2E are shown using circles and theexact values are shown as a con^nuous black line. We use Tro^er steps ^^ = 0.1 to implementthe ^me evolu^on. There are 1000 shots per circuit.Canonical ensemble expecta^on values Finally, we show that using a shuffling quantum channel with mul^ple ancillas, one can compute expecta^on values atfinite temperature. Although this applica^on suffers from exponen^alcomplexity and consequent system size and / or run^me requirements, the applica^on shows theversa^lity of our channel in extrac^ng physical informa^on about a system. This can be expressedmore generally, as follows We saw previously that the output of the shuffling channel can beexpressed with two coefficients A and B, with This expression is reminiscent ofa par^^on func^on. As a comparison, expecta^on values atfinite inverse temperature ^ read^^ [^^^^^]. If the eigenval ^ ^^^^^ [^^^^] ues of H are close to 0, the term cos (^) will be close to ^ forsome appropriate value of ^. If we could implement higher powers cos^^(^) with N integer,we could make this term arbitrarily close to ^^^^up to propor^onality factors. It turns out that these higher powers can be implemented by generalizing the shuffling channel to mul^ple ancillas.To be more specific, let us consider a Hamiltonian ^ on ^ qubits, an observable O, an ancillaqubit ^ and an inverse temperature ^. We defineA^orney ref: 65193 PCT^′ = arctan^ +^^^^^^^ ^^^, (27)on the system of ^ qubits plus one ancilla qubit ^ , where ^^denotes the Pauli matrix ^applied on the ancilla ^, and where we introduced parameters ^ > 0 and ^ > 0, as well as aninteger ^. The addi^onal ancilla ^ is not the ancilla of the shuffling channel. When applying theshuffling channel as described below, this ancilla ^ will be considered part of the system. Theparameters are chosen so that cos^^ converges to a Boltzmann weight. Namely, for ^ larger than the norm of ^^ / ^, we have

[0050] : Now, as shown in Figure 6 and described in the sec^on ‘Generaliza^on of the channel to mul^pleancillas’ described below, the shuffling quantum channel can be generalized into a channelShuffle(^, ^) that uses ^ ancillas as follows: one applies step 2 of the method implementedby the channel to each of the ^ ancillas, and then in step 3 one applies ^^ … ^^ on the ^qubits only if all the ^ ancillas have been measured to be in state 0. An iden^cal rela^on to (2)follows then with precise coefficients ^ and ^ given in the sec^on ‘Generaliza^on of thechannel to mul^ple ancillas’. In the par^cular case of ^′ in (27), because ^^ commutes with^′, we have t^ [^′^^] = ^t^ [^^^] + ^ , (29)with where T^ denotes the trace on the ^ qubits and the ancilla ^ . Hence, by repea^ng thisgeneralized channel we obtain ^→^ 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 ^ we obtain in this regime where we now use t^ to denote the trace over the ^ qubits, without the qubit ^ .Interes^ngly, the algorithm automa^cally computes the ra^o of traces t^ [^^^^^] / 2^ andt^ [^^^^] / 2^ that can each of them be exponen^ally small.Using (28), (29) and (30), the number of rounds ^ required to reach this limit is 1 / log(1 −^^ [^ ^^^]^^(^^^^)^). The ^me complexity is thus exponen^al in the system size ^, which is the expectedA^orney ref: 65193 PCTscaling for afinite-temperature algorithm that applies to generic Hamiltonians. However, thisapplica^on shows the versa^lity of our shuffling channel to prepare qubits in a specific state. DiscussionThis patent specifica^on provides evidence for the usefulness of mid-circuit measurements andclassical condi^oning on the measurement outcomes (also called feed-forward) as integralelements of quantum algorithms that achieve built-in noise resilience, and not only as quantumerror correc^on subrou^nes. We introduced a “shuffling" quantum channel based on mid-circuit measurements and feed-forward with two verified proper^es. Firstly, it can be used to compute dynamical correla^ons at infinite temperature andfinite temperature expecta^on values for any Hamiltonian. Although the algorithm forfinite temperature expecta^on values is exponen^al in system size run^me, the applica^on to dynamical correla^ons at infinite temperature is ofcurrent prac^cal use. Secondly, it displays a noise-limi^ng behaviour, in the sense that it outputsa non-zero signal on a noisy hardware even when repeated an arbitrary number of ^mes. We give analy^cal, numerical and experimental evidence on the H1-1 trapped-ion quantum computer from Quan^nuum for this noise-limi^ng property. This shuffling quantum channel shows that algorithms based on mid-circuit measurements and feed-forward can support higher levels of noise than purely unitary circuits, making them par^cularly interes^ng for present-day quantum computers.In this specifica^on, we have provided afirst example of how non-error-corrected algorithms canavoid total depolarisa^on in the infinite-depth limit. Furthermore, although the algorithms presented here forfinite temperature expecta^on values have an exponen^al ^me complexity(and this is unavoidable without further assump^on on ^ ), the inventors have noted thepoten^al for improvements for local Hamiltonians ^. Generaliza^on of the channel to mul^ple ancillas In this Sec^on we generalize the shuffling quantum channel to mul^ple ancillas. Namely, for aninteger ^ we define Shuffle(^, ^) through the following steps.1. Pick a Pauli matrix at each site, i.e. ^^, ... , ^^ ∈ {^, ^, ^, ^} uniformly at random.2. Prepare ^ ancillas in state |0〉^^ ... |0〉^^. For ℓ = 1, ... , ^, apply ^^^^^...^^ ^^ℓ. 3. Measure the ^ ancillas. Only if the measurement outcome is 0 for all the ^ancillas, apply ^^... ^^ on the ^ qubits.The analysis of Shuffle(^) is straigh^orwardly generalized to the case ^ > 1. For any Paulistring operator ^ = ^′^... ^′^ with ^′^ ∈ {^, ^, ^, ^}, we haveT^ [^^′] = ^T^ [^^] + ^ , (33)with ×(^) (^)] ,andA^orney ref: 65193 PCT Moreover, the probability ^ of measuring 0 in all the ^ ancillas is (36)When ^ commutes with ^, we obtain As noted above, the system on which we apply this quantum channel contains ^ qubits and oneextra ancilla ^. It is thus a system with ^ + 1 qubits. Moreover the Hamiltonian ^′ commuteswith the par^cular choice of Pauli string ^ = ^^ that we consider. This yields thus formula (30)above.Alterna^ve way of extrac^ng the coefficients ^ and ^In this Sec^on we present an alterna^ve way of compu^ng coefficients ^ and ^ appearing inthe shuffling quantum channel output in Equa^on (2) above. By applying twice Shuffle(^) on adensity matrix ^, and deno^ng ^′′ the output density matrix, we have for any Pauli string ^T^ [^′′^] = ^^T^ [^^] + ^^ + ^ . (38)It follows thus that we can extract ^ from the ra^o where we recall that ^′ denotes the density matrix a^er one single run of Shuffle(^). Then,^ is given by^ = T^ [^′^] − ^T^ [^^] . (40)Applying this process to the Hamiltonians ^^ and ^^ +^ ^, one can then sum the resul^ng ^’s and obtain the dynamical correla^on ^^(^). This way of compu^ng ^^(^) presents some^mes a lower variance compared to the onequoted in the main text, although more involved. Let us indeed compute the variance Δ in thees^ma^on of ^. The variance in es^ma^ng T^ [^′^] using ^ shots is ^ . 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Claims

A^orney ref: 65193 PCTClaims1. A computer-implemented computa^on method comprising:genera^ng a quantum channel, including opera^ons to be performed with respect toquantum states, to represent the evolu^on of a physical system, wherein the quantum channelcomprises quantum circuits adapted to perform at least afirst computa^on step on a set of qubitsincluding 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 computa^on step that is condi^onal on a result of the mid-circuit state measurement, to generate a mixed state output of the quantum channel; and execu^ng the at least one quantum channel on a set of qubits of a quantum compu^ng apparatus, including performing at least afirst computa^on 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 condi^onal computa^on step on a subset of the qubits otherthan the measured ancilla qubit, thereby to generate a mixed state output containinginforma^on rela^ng to the physical system; andwherein the mixed state output is input to a subsequent execu^on of the quantum channel, and wherein execu^on of the quantum channel is repeated.

2. A method according to claim 1, wherein the computer system comprises a classical compu^ng apparatus and a quantum compu^ng apparatus, with the genera^on of the quantum channel performed on the classical compu^ng apparatus and the execu^on of the quantum channel performed on the quantum compu^ng 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 afirst computa^on step establishes a correla^on 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 thefirst computa^on on the other qubits.

67. A method according to claim 1, wherein execu^on of the quantum channel is repeated mul^ple ^mes un^l the output mixed state has converged to afixed point of the quantum channel.A^orney ref: 65193 PCT7. A method according to claim 1, wherein the qubits of the set of qubits are prepared in statesdefined in probability density matrix represen^ng possible states of the physical system prior toperforming thefirst computa^on step, and execu^on of the quantum channel outputs a modified probability density matrix represen^ng possible states of physical system, wherein the output matrix converges to an equilibrium state of the quantum channel a^er repeated execu^on of the quantum channel.

8. A method according to any preceding claim, wherein the set of qubits of the quantum compu^ng apparatus are implemented using trapped ions 9. A method according to any preceding claim, wherein the genera^on of a quantum channel is performed in response to an input representa^on of a physical system for which the computa^on method is required.

10. A method according to claim 9, wherein the representa^on of the physical system is aHamiltonian.

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

12. A method according to any preceding claim, wherein the representa^on of a physicalsystem is an Hamiltonian or other mathema^cal representa^on of a quantum physical system and the quantum channel is adapted to calculate expecta^on values or dynamical correla^onsfor operators and / or quantum states of a quantum physical system.

13. A quantum compu^ng system comprising: afirst compu^ng apparatus configured to generate a quantum channel for performing a sequence of computa^onal steps for compu^ng informa^on rela^ng to a physical system, the quantum channel comprising quantum circuits adapted to perform at least afirst computa^on 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 computa^on step that is condi^onal 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 compu^ng apparatus comprising a set of qubits for execu^ng the quantumchannel, wherein the quantum compu^ng apparatus is configured to perform at least afirstA^orney ref: 65193 PCTcomputa^on 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 one condi^onal computa^on step on a subset of the qubits other than the measured ancilla qubit, thereby to generate a mixed state output containing informa^on rela^ng to the physical system, and wherein the mixed state output is input to a subsequent execu^on of the quantum channel, and wherein execu^on of the quantum channel is repeated.

14. A system according to claim 13, wherein the quantum compu^ng apparatus is configured to repeat execu^on of the quantum channel using the mixed state output has converged to afixed point of the quantum channel.

15. A system according to claim 13, wherein the set of qubits of the quantum compu^ng apparatus are implemented using trapped ions 16. A system according to claim 13, wherein the quantum compu^ng apparatus includes L computa^onal qubits and one or more ancilla qubits, and the quantum channel is configured to perform at least afirst opera^on on afirst 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 condi^onal computa^on step on the L qubits excluding the measured ancilla qubit(s).

17. A system according to claim 16, wherein a Pauli matrix is allocated to each of the L computa^onal qubits, and the one or more ancilla qubits are ini^alized in afirst state, and thenafirst opera^on is performed on the L qubits and the one or more ancilla qubits based on theallocated Pauli matrix, and wherein a mid-circuit measurement is performed on the one or more ancilla qubits, and wherein the respec^ve Pauli matrix is applied to each of the L computa^onal qubits on condi^on of a specific measurement result of the one or more ancillas qubits.

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