Method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor, computer program product, data carrier and computing system

Abstract

The invention is related to a method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor to a computer program product and to a computing system for carrying out the method and to a data carrier having stored thereon the computer program.

Description

[0001] Method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor, computer program product,

[0002] data carrier and computing system

[0003] The invention is related to a method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor, to a computer program product and a computing system for carrying out the method and to a data carrier having stored thereon the computer program.

[0004] In recent years significant progress has been made in improving qubit coherence and increasing qubit counts across various quantum computing platforms. However, the pursuit of achieving fault-tolerant quantum computation through quantum error correction remains a challenge. In the quest for gaining quantum advantage in the Noisy Intermediate Scale Quantum (NISQ) processor domain, the development of quantum error mitigation techniques has garnered significant interest in recent years, see e.g. S. Endo et.al, “Hybrid quantum- classical algorithms and quantum error mitigation”, Journal of the Physical Society of Japan 90, 032001 (2021), Z. Cai et.al, “Quantum error mitigation”, (2023), arXiv: 2210.00921, Y. Quek et.al, “Exponentially tighter bounds on limitations of quantum error mitigation”, (2023), arXiv: 2210.11505. The general idea here is to devise strategies for understanding and compensating for the effects of hardware noise on the outcomes of quantum algorithm executions, particularly those used to estimate mean values / expectation values.

[0005] Numerous error mitigation techniques (for mean value estimation) have emerged in the past, and they can be broadly categorized into two distinct branches: noise-agnostic and noise-aware methods. In the noise-aware category, and particularly for Probabilistic Error Mitigation (PEC), a precise model of the underlying hardware noise is essential to effectively neutralize its impact; see e.g. K. Temme et.al, “Error mitigation for short-depth quantum circuits,” Phys. Rev. Lett. 119, 180509 (2017), S. Endo et.al, “Practical quantum error mitigation for near-future applications”, Phys. Rev. X 8, 031027 (2018), C. Song et.al, “Quantum computation with universal error mitigation on a superconducting quantum processor,” Science Advances 5, eaaw5686 (2019). While capable of entirely removing the noise’s impact on the expectation value, PEC comes at the cost of an exponential sampling overhead. In addition and despite good progress in achieving scalable error characterization techniques (see e.g. E. van den Berg et.al, “Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors”, Nature Physics 19, 1116-1121 (2023)), it is crucial to recognize that achieving a perfectly accurate noise model (including spatially and temporally correlated

[0006] P00183 WO 05122025 errors) might practically be an impossible task, particularly when dealing with more than just few qubits.

[0007] Noise-agnostic error mitigation is the other distinct branch in which having a noise model for the quantum hardware is not required, and it is generally known to have smaller sampling overhead compared with PEC (still with exponential scaling albeit with a smaller base). Here one tries to understand the effect of the noise on the expectation value of the observable of interest in order to find (an approximation for) the noiseless expectation value. While different noise-agnostic error mitigation methods work based on different principles and each technique has its own shortcomings, traditionally many of them (as briefly reviewed below) suffer from a common bottleneck: they rely on fitting the data to a specific mathematical function. This in turn introduces a systematic unknown bias to the outcome of the error mitigation.

[0008] A well-known example of noise-agnostic error mitigation is Zero Noise Extrapolation (ZNE) as introduced in K. Temme et.al, Phys. Rev. Lett. 119, 180509 (2017) cited above and Y. Li et.al, “Efficient variational quantum simulator incorporating active error minimization”, Phys. Rev. X 7, 021050 (2017). Here, understanding the effects of the noise is performed by amplifying the noise and measuring the observable of interest at various noise scales. The set of noisy expectation values is then extrapolated to the limit in which the noise scale is zero. The extrapolation method is commonly based on either the Richardson technique (which uses a polynomial fit, see the above-cited references K. Temme et.al, Phys. Rev. Lett. 119, 180509, Y. Li et al., Phys. Rev. X 7, 021050, and M. Krebsbach et.al, “Optimization of Richardson extrapolation for quantum error mitigation”, Phys. Rev. A 106, 062436) or a single-exponential fit (see S. Endo et.al, Phys. Rev. X 8, 031027 cited above or Y. Kim et.al, “Evidence for the utility of quantum computing before fault tolerance”, Nature 618, 500-505 (2023)). Nevertheless, it is known that for a quantum circuit including non-Clifford gates and in the presence of Pauli noise, the exact functional form to fit an expectation value of any Pauli observable shall be a multi-exponential function containing a sum of an unknown number of different single exponential functions (see Y. Kim et.al, Nature 618 cited above or Z. Cai “Multiexponential error extrapolation and combining error mitigation techniques for NISQ applications”, npj Quantum Information 7, 80 (2021)).

[0009] In addition, in ZNE a precise noise amplification is required. Any inaccuracy in the noise amplification, in turn, contributes to the eventual bias error. The noise amplification can be performed either by pulse-level control (see K. Temme et al., Phys. Rev. Lett. 119 cited above or A. Kandala et.al, “Error mitigation extends the computational reach of a noisy quantum processor”, Nature 567, 491-495 (2019)), or gate-level control, using either unitary folding (see

[0010] P-00183WO 05122025 T. Giurgica-Tiron et.al, “Digital zero noise extrapolation for quantum error mitigation”, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, 2020)) or local folding (see E. F. Dumitrescu et.al, “Cloud quantum computing of an atomic nucleus”, Phys. Rev. Lett. 120, 210501 (2018), A. He et.al, “Zero-noise extrapolation for quantum-gate error mitigation with identity insertions”, Phys. Rev. A 102, 012426 (2020)). It has been shown that unitary folding can be more reliable compared with local folding (K. Schultz et.al, “Impact of time-correlated noise on zero-noise extrapolation”, Phys. Rev. A 106, 052406 (2022)). In either case, the gate-level noise-amplification techniques perform the best once the intended noise scaling factor is an odd-integer number. Even in this case, it has been shown that the true noise scale factors can differ from the intended ones due to the fact that the noisy quantum channel describing the hardware noise may not commute with the ideal circuit unitary (see K. F. Koenig et.al, “Inverted-circuit zero-noise extrapolation for quantum gate error mitigation”, (2024) arXiv: 2403.01608). More accurate noise amplification was recently achieved in a probabilistic method at the cost of having a detailed noise model, see Y. Kim et.al, Nature 618 cited above, S. Ferracin et.al, “Efficiently improving the performance of noisy quantum computers”, (2022), arXiv: 2201.10672, L. Hour et.al, “Improving zero-noise extrapolation for quantum-gate error mitigation using a noise-aware folding method”, (2024), arXiv: 2401.12495, (in which case the resulting error mitigation can no longer be considered a noise-agnostic method.)

[0011] In another method, see M. Urbanek et al., “Mitigating depolarizing noise on a quantum computer with noise-estimation circuits”, Phys. Rev. Lett. 127, 270502 (2021), the authors introduce using an estimation circuit, obtained by removing all single qubit-gates of the target circuit, in order to find an estimation of the depolarizing rate influencing the target circuit. Once this rate is approximately found, the authors consider the depolarizing-error mitigated expectation value and use this quantity as the input for ZNE. Provided that depolarizing noise is the dominant error mechanism, the authors argue that their technique can yield more accurate results compared with the standard ZNE. It was left as an open question in this reference how to generate better estimation circuits, and whether the technique remains valid when the depolarizing noise is not the dominant error. In addition, since this technique relies on ZNE, accurate noise-amplification is required.

[0012] Another example of noise-agnostic error mitigation is Clifford Data Regression (CDR), see P. Czarnik et.al “Error mitigation with Clifford quantum-circuit data”, Quantum 5, 592 (2021), and its extended model known as variable-noise Clifford Data Regression (vnCDR), see A. Lowe et.al, “Unified approach to data-driven quantum error mitigation”, Phys. Rev. Res.

[0013] 3, 033098 (2021 ). In these methods, many near-Clifford circuits are generated from the original

[0014] P-00183WO 05122025 target quantum circuit. The expectation value of the observable of interest is then found by measuring the noisy near-Clifford circuits as well as simulating them in a noise-free environment. This provides a set of training data from which one can learn (e.g., via linear regression in CDR) how the noise influences each of the simulated expectation values. However, due to generally different noise-scaling behavior between the near-Clifford circuits and non-Clifford circuits, the model learned from near-Clifford circuits is expected to become less accurate as the depth of the target non-Clifford circuit increases.

[0015] Due to these problems in the prior art, it is therefore an object of the present invention to provide a method for quantum error mitigation with improved efficiency and accuracy.

[0016] According to a first aspect of the present invention, this object is attained by a method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor on the basis of a target set associated with the target quantum circuit, a plurality of auxiliary sets, each being associated with a distinct auxiliary quantum circuit of a plurality of auxiliary quantum circuits each being different from the target quantum circuit, and for each of the auxiliary quantum circuits an associated noise-free auxiliary value, said target set comprising for each of a plurality of target noise strength values an associated target estimator value for an expectation value of an observable, the target estimator values being determined by evaluation of target measurement data obtained by an execution of the target quantum circuit by the quantum processor at a plurality of different target noise levels, each target noise level being associated with one of the target noise strength values, each auxiliary set comprising for each of a plurality of auxiliary noise strength values an associated auxiliary estimator value for an expectation value of the observable, the auxiliary estimator values being determined by evaluation of associated auxiliary measurement data obtained by an execution of the auxiliary quantum circuit associated with the auxiliary set by the quantum processor at the plurality of different auxiliary noise levels, each auxiliary noise level being associated with one of the auxiliary noise strength values, and each noise-free auxiliary value being an expectation value of the observable for a fictitious noise-free execution of the associated auxiliary quantum circuit or an estimator value thereof, said method comprising:

[0017] for each of the auxiliary quantum circuits, executing a routine for inferring a first error-mitigated target estimator value associated with the respective auxiliary quantum circuit, the routine comprising:

[0018] a noise-reduction step of deriving for each target noise strength value a noise-reduced target estimator value associated with the auxiliary quantum circuit, said noise-reduced target estimator value comprising first and second contributions, wherein the first contribution is derived by reducing a noise-dependency of the target estimator values on the basis of the

[0019] P-00183WO 05122025 auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit, and the second contribution is configured to further reduce a remaining noisedependency of the first contribution and is derived by use of a control parameter-dependent model for the remaining noise dependency on the basis of the auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit;

[0020] a control parameter step of determining an optimal value of the control parameter such that the noise-dependency of the noise-reduced target estimator values fulfills a minimization criterion;

[0021] an inference step of inferring the first error-mitigated target estimator value associated with the auxiliary quantum circuit from the noise-reduced target estimator values with the optimal control parameter value associated with the auxiliary quantum circuit;

[0022] said method further comprising:

[0023] inferring a final error-mitigated target estimator value on the basis of an inference set which comprises the plurality of first error mitigated target estimator values.

[0024] At least some of the steps of the method according to the first aspect of the present invention may be implemented on a computer, in particular on a classical computer in one example. In one example, all steps may be implemented on a classical computer. In this computer-implemented method, in particular the routine for inferring the first error-mitigated target estimator value may be executed by a classical computer, and inferring the final error- mitigated target estimator value may be by the classical computer. In some realizations, at least one step may be implemented by quantum processing.

[0025] The wording used throughout this application is in line with standard terminology used in the field of quantum computation, see, e.g., the standard textbook M. Nielsen, I. Chuang, “Quantum Computation and Quantum Information”.

[0026] A quantum processor may be understood as a computer that exploits the laws of quantum mechanics. In particular, the quantum processor may comprise a plurality of N>2 quantum mechanical d-level systems, d≥2, (also known as qudits) used as the carriers of information. The quantum mechanical d-level systems are physical objects realized, e.g., in superconducting systems, photonic systems, ion systems, etc., but the invention is not limited to this. In particular, for d=2 the quantum mechanical d-level systems are qubits. For ease of representation, the present specification explains the case of qubits. However, everything said within this specification equally applies to qudits with d>2.

[0027] P-00183WO 05122025 The qubits may be described by the mathematical formalism of quantum mechanics. There, each qubit is associated with a two-dimensional Hilbert space. Throughout this specification, we denote the two basis vectors of the Hilbert space associated with each qubit by |0> and |1>. We may also refer to these basis states as computational basis states.

[0028] The quantum processor may further comprise state preparation means configured to prepare the qubits in a predetermined initial state. In one expedient example, the initial state is of the form |00...0>, that is, all qubits are in the state |0>. However, the invention is not limited to this. The quantum processor may further comprise means configured to apply at least one quantum gate, and preferably a plurality of different quantum gates, to the N qubits. Within the mathematical formalism of quantum mechanics, a quantum gate may be understood as a unitary transformation that acts on a state of a small number of qubits. In one example, the quantum processor may be configured to apply at least one single-qubit gate and at least one two-qubit gate. In one example, the quantum processor may be a universal quantum processor. That is, the quantum processor is configured to realize any unitary transformation defined on N qubits by an application of a sequence of quantum gates.

[0029] The quantum processor may further comprise measurement means for applying a quantum measurement to the N qubits to thereby obtain measurement data. The measurement data is classical data. In one example, the quantum measurement may be a quantum measurement in the computational basis. In another example, the quantum measurement may be a measurement in the local Pauli basis of each qubit.

[0030] The quantum processor may be part of a computing system which may also comprise a classical computer configured for control of the quantum processor and / or for classical postprocessing of the measurement data. The classical computer may be understood as a computer that is based on classical physics. In particular, the classical computer may be configured to apply operations, in particular binary logic gates, to classical bits.

[0031] The observable is related to a physical property of the system of the N qubits. The observable may, e.g. be the energy of a physical system encoded in the N qubits or an occupation number of one of the states |0> and 11 > for each qubit, but the invention is not limited to this. Within the mathematical theory of quantum mechanics, the observable is associated with a Hermitian operator. Due to the probabilistic nature of quantum mechanics, only an expectation value of the observable may be determined. In an experiment with finite statistics, only an estimator value of the expectation value of the observable may be determined. In one example, the estimator value may be a mean value. The quantum

[0032] P-00183WO 05122025 measurement is such that the obtained measurement data allows to determine the estimator value of the observable from the measurement data.

[0033] A quantum circuit executed by the quantum processor comprises instructions on the N qubits. In particular, the quantum circuit comprises instructions which specify an initial state of the N qubits which may be understood as a starting point of the computation. In one example, the initial state may be the state described by |00...0>. The quantum circuit further comprises instructions specifying a gate sequence of quantum gates which, in the noiseless case, realize a unitary transformation on the N qubits. In an expedient example, each quantum gate of the sequence of quantum gates may be a single-qubit gate or a two-qubit gate. The quantum circuit further comprises an instruction to apply a quantum measurement on the N qubits. In one example, the quantum measurement is the measurement in the computational basis of the N qubits.

[0034] Due to the probabilistic nature of quantum mechanics, an execution of the quantum circuit comprises an implementation of a number of S shots of the quantum circuit A shot of the quantum circuit is an implementation of the instructions of the quantum circuit as explained above, i.e. state preparation of the initial state, application of the quantum gate sequence and quantum measurement. Each shot s results in an associate measurement outcome of the quantum measurement. The measurement data comprises all measurement outcomes. The measurement data may allow to calculate the estimator value of the observable 0. In one example the estimator value may be the mean value of the observable. In one example, S is at least 104, preferably at least 105, more preferably at least 106. In another example, S may be less than 104.

[0035] The execution of any quantum circuit on a quantum processor is inevitably noisy. The noise level, i.e. the strength of the noise, may be associated with a noise strength value (also known as noise scale factor in the art) A. A value of A - 1 may be associated with the base hardware noise. That is, A - 1 is associated with the least noisy execution of the quantum circuit on the respective quantum processor. Therefore, all noise strength values may fulfill A a 1. In one example, the noise strength values increase with the noise level. As it is known in the art of quantum error mitigation, there are various noise scaling techniques to increase the noise level and thereby the noise strength value A of an execution of a quantum circuit. One example is unitary folding, wherein the target (auxiliary) gate sequence of the target (auxiliary) quantum circuit is supplemented with further quantum gates so that the total unitary transformation realized by the supplemented gate sequence is equal to the target (auxiliary) gate sequence in the noiseless case. This may be achieved, for example, by inserting a quantum gate and its

[0036] P-00183WO 05122025 inverse into the target (auxiliary) gate sequence. The implementation of the supplemented quantum gates introduces additional noise in the execution of the quantum gate sequence. For example, when the quantum gate sequence is described by a unitary U, noise strength values λm = 2m + 1, m = 1,...,M may be achieved by an application of a supplemented gate sequence described by a unitary (

[0037]

[0038] UU^)mU. However, the invention is not limited to unitary folding, and other possibilities of executing quantum circuits at various noise strength values will be explained below. In particular, the target (auxiliary) noise strength values A are always at least 1 (base hardware noise) and may in particular have an integer or a rational value, though it is not limited to this. Furthermore, within this specification, we denote the plurality of different target noise strength values associated with the plurality of different target noise levels as A® < ° < - <, M > 3, and we denote the plurality of different auxiliary noise strength values associated with the plurality of different auxiliary noise levels as A^ < A^ < ••• < A^), M' > 3.. In one example, A® = 1 and / or A^ = 1. The difference between two consecutive noise strength values, 2^'“^ and 2^'“-’ is also denoted as spacing between these noise strength values. The spacing may be uniform for the noise-strength values of the plurality in one example. In another example, the spacing may be different for at least two pairs of consecutive noise strength values. The number of different target and auxiliary noise levels, i.e., the number of different noise strength values, may be M = 3, 4, 5 or more and / or M’ - 3, 4, 5 or more in one example. More noise strength values are computationally more costly, but may result in a higher accuracy of the error-mitigated target estimator value.

[0039] In one expedient example, the number of different target noise strength values is equal to the number of different auxiliary noise strength values. In one example, the target and auxiliary noise strength values are equal in their number and their values, i.e. λ^(m) =

[0040]

[0041] for each m = 1,..., M. However, the invention is not limited to this. In certain embodiments it may also be envisioned that the number and / or values of the auxiliary noise strength values are different for at least two auxiliary quantum circuits. The target quantum circuit is a quantum circuit of interest. For example, the target quantum circuit may solve a particular computational problem of interest. In particular, the computational problem may not be solvable efficiently on a classical computer, but the target quantum circuit may allow for an efficient solution on the quantum processor. The target quantum circuit is defined on the N≥2 qubits of the quantum processor. The target quantum circuit comprises instructions to apply a target gate sequence to an initial target state of the N qubits to thereby obtain a final target state. A target measurement is applied to the final target state to thereby obtain target measurement data.

[0042] P-00183WO 05122025 Each of the plurality of J>2 auxiliary quantum circuits Aj, j=1, J is defined on the same number N of qubits as the target quantum circuit. In one example, J is at least 20, preferably at least 50, more preferably at least 100. The auxiliary quantum circuits are all different from each other. Each auxiliary quantum circuit Aj comprises instructions to apply an auxiliary gate sequence to an initial auxiliary state of the N qubits to thereby obtain a final auxiliary state. An auxiliary measurement is applied to the auxiliary final state to thereby obtain the auxiliary measurement data. In one expedient example, the initial auxiliary state is the same for all auxiliary quantum circuits. In one expedient example, the initial auxiliary state of each auxiliary quantum circuit is identical to the initial target state. Additionally, or alternatively, the auxiliary measurement of the plurality of auxiliary quantum circuits may all be identical in one example. In one expedient example, the auxiliary measurements may be identical to the target measurement for each auxiliary quantum circuit.

[0043] The auxiliary gate sequence of each auxiliary quantum circuit may be derived from the target gate sequence in one example. In particular, each auxiliary gate sequence may be different from the target gate sequence. The auxiliary gate sequence of each auxiliary quantum circuit and the target gate sequence may be such that they have certain quantum gates in common but that they differ in other quantum gates of their respective quantum gate sequences in one example. In one example, each auxiliary quantum circuit may be such that the final auxiliary state of the fictitious noiseless execution of the auxiliary quantum circuit, in particular of the noiseless implementation of the auxiliary gate sequence, is known or may be obtained in an efficient way. In one example, the execution of at least one, and preferably of each, auxiliary quantum circuit may be efficiently classically simulable. In particular, a quantum circuit is efficiently classically simulable when its runtime scales less than exponential in the number of qubits, and in particular linear in the number of qubits. In particular, at least one, and preferably each, auxiliary quantum gate sequence may comprise only Clifford gates as the auxiliary quantum circuit is efficiently classically simulable in this case. (This is also known as Gottesman-Knill theorem; see also D. Gottesman, “The Heisenberg Representation of Quantum Computers”, arXiv:quant-ph / 9807006v1.) For example, at least one, and preferably each, auxiliary gate sequence may be derived from the target gate sequence by replacing each non-Clifford gate of the target gate sequence by some Clifford gate. In another example, at least one, and preferably each, auxiliary quantum gate sequence may be such that the final auxiliary state obtained by an application of the auxiliary quantum gate sequence to the auxiliary initial state is known. For example, at least one, and preferably each, auxiliary quantum gate sequence may correspond to a unitary transformation which has a simple or trivial action on the auxiliary initial state, for example the auxiliary gate sequence may be equivalent to a unitary transformation which is the identity.

[0044] P-00183WO 05122025 According to the method of the first aspect of the present invention, the target set comprises for each of the plurality of target noise strength values 2® < 2® < ••• < 2^, M>3, an associated target estimator value for the expectation value of the observable. The target set may comprise tuples (A®, E(A^, T)) wherein E(λ^(m), T)denotes the target estimator value for the expectation value of the observable obtained by the execution of the target quantum circuit with the target noise strength value A®. That is, the target set comprises classical data. The target set may be provided as an input to the method or it may be derived as part of the method as it is further explained below.

[0045] In a similar way, each auxiliary set associated with one of the auxiliary quantum circuits Aj comprises for each of the plurality of auxiliary noise-strength values an associated auxiliary estimator value for the expectation value of the observable. That is, the auxiliary set associated with the auxiliary quantum circuit Aj may comprise tuples (λ^(m) E (λ^(m) Aj)) wherein E (λ^(m) Aj) denotes the auxiliary estimator value for the expectation value of the observable obtained by the execution of the auxiliary quantum circuit Aj with the noise-strength value

[0046]

[0047] Similar to the target set, each auxiliary set comprises classical data. The plurality of auxiliary sets may be provided as an input to the method or it may be derived as part of the method as it is further explained below.

[0048] For each auxiliary quantum circuit Aj the associated noise-free auxiliary value is the expectation value of the observable for a final state of the (fictitious) noiseless execution of the auxiliary quantum circuit Aj or an estimator value thereof. The noise-free auxiliary value is denoted as E(0, Aj) throughout this specification. In certain cases, e.g., when the final auxiliary state of the noiseless execution of the auxiliary quantum circuit Aj is known, the noise-free auxiliary value may be the exact expectation value. In other instances, e.g., when the final auxiliary state of the noiseless execution of the auxiliary quantum circuit is obtained by a classical simulation of the noiseless execution of the auxiliary quantum circuit, the noiseless auxiliary value may be an estimator value of the expectation value, or it may be the exact expectation value. -

[0049] The method comprises data processing of the target set, the plurality of auxiliary sets and the plurality of noise-free auxiliary values. In particular, the method comprises for each of the auxiliary quantum circuits Aj an execution of a routine for inferring a first error-mitigated target estimator value associated with the respective auxiliary quantum circuit Aj. The routine comprises for each auxiliary quantum circuit Aj a noise-reduction step, a control parameter P-00183WO 05122025 step and an inference step. These steps are explained below for the auxiliary quantum circuitAj.

[0050] In the noise reduction step a noise-reduced target estimator value is derived for each target noise strength value. The deriving is such that the set of the derived noise-reduced target estimator values has a reduced noise-dependency compared to the target estimator values of the target set. For example, a value of a measure of dispersion for the plurality of noise-reduced target estimator values may be smaller than a value of the measure of dispersion for the plurality of target estimator values.

[0051] The deriving of the noise-reduced target estimator values is such that each noise- reduced target estimator value associated with one of the target noise strength values comprises first and second contributions. In one expedient example, each noise-reduced target estimator value may comprise only the first and second contributions and no further contribution. The first contribution is derived by reducing the noise-dependency of the target estimator values on the basis of the noise-dependency of the auxiliary estimator values EtA^, Aj) and the noise-free auxiliary value E(0, Aj)) associated with the auxiliary quantum circuit Aj. For example, one may assume that the target estimator values and the auxiliary estimator values have a similar noise dependency. Specific examples how the noise dependency of the target estimator values may be reduced are presented below. The first contribution for the auxiliary quantum circuit Aj is denoted asfJ within this

[0052]

[0053] specification. Here, the notationtJ means that for each target

[0054]

[0055] estimator value A® the first contribution depends on the target estimator value E(A^, T) for the target noise strength value A® and on all auxiliary estimator valuestassociated with the auxiliary quantum circuit Aj or a subset thereof. Deriving

[0056]

[0057] noise-reduced target estimator values having only the first contribution is known in the art.

[0058] In general, the first contribution still has some noise dependency as the target quantum circuit and the auxiliary quantum circuit Aj differ and therefore their noise behavior will in general not be identical. Therefore, a control parameter-dependent model is used to further reduce the remaining noise dependency of the first contribution. The control parameter is in particular a real-valued parameter. The model may be derived on the basis of well-thought considerations. Possible examples of such models will be presented below. In one expedient example, the control parameter-dependent model may be linear in the control parameter. The

[0059] P-00183WO 05122025 second contribution is denoted by P2( A®, |E(2^?, J47)}t, E(o, Aj),nj), wherein nj is the

[0060]

[0061] control parameter associated with the auxiliary quantum circuit Aj. In particular, the control parameter may be a single parameter, and in particular a single real-valued parameter, in one example, but the invention is not limited to this. In another example, the control parameter may comprise a plurality of control parameters, and preferably a plurality of real-valued control parameters. Here, the notation P2( λ^(m){E(2^?,. / 17)}(, E(0, Aj)),n7) means that for each

[0062]

[0063] target estimator value A® the second contribution depends on the noise-free auxiliary estimator value associated with the auxiliary quantum circuit Aj, the control parameter n7- and on all auxiliary estimator values ^E(2^?,47-)j(associated with the auxiliary quantum circuit Aj or a subset thereof.

[0064] In one expedient example, the noise-reduced target estimator value for the target noise strength value A® may be of the following form:

[0065] P2(*S?,. £(0. A, ). nt).

[0066]

[0067] For each target noise strength value, the first and second contributions may depend on all auxiliary estimator values or a true subset thereof. The true subset may be different for different target noise strength values in one example. In one example where the number of target and auxiliary noise strength values is identical, 2^ ~

[0068]

[0069] ) f°rsome bijective function G. In this case, the value of the first and second contributions for the target noise strength value 2®maYonly depend on the associated auxiliary noise strength value 2^ = G(2®) in one example. I.e. EredQ^,nj, Aj) = P^E^lTlECGCA^. Aj)) + P2Q^, E(G(A(^), A), E(0, Aj)'),nj). In one expedient example, 2^)= G(2®) = x2® for some positive x > 0. In particular, x-1. Then, Ered(λ^(m)n7, Aj) - P1(EQ^, T), E(xA^, Aj) )+ P

[0070]

[0071] 20^, E(XA^, Aj), E(0, Aj)),nj).

[0072] The noise-reduced target estimator values derived in the noise reduction step depend on the value of the control parameter nj.

[0073] P-00183WO 05122025 The method further comprises a control parameter step of determining an optimal value of the control parameter n7<opassociated with the auxiliary quantum circuit Aj such that the noise-dependency of the noise-reduced target estimator values fulfils the minimization criterion. For example, the optimal value of the control parameter may be selected such that a value of a measure of dispersion of the noise-reduced target estimator values is below a threshold value. In particular, the optimal value of the control parameter may be determined such that the value of the measure of dispersion is minimal. In one particular example, the value of the control parameter may be selected such that the value of the measure of dispersion is zero.

[0074] Once the optimal value of the control parameter is determined, a first error-mitigated target estimator value E^Aj) is determined on the basis of the noise-reduced target estimator values with the optimal control parameter value EredQ^,njOp, Aj) in the inference step.

[0075] In some examples, the noise-reduction step and / or the control parameter step and / or the inference step may be performed at least partially, and in particular completely, on a classical computer.

[0076] Once the routine has been executed for each of the plurality of auxiliary quantum circuits Aj, j=1,..., J, the method proceeds with inferring the final error-mitigated target estimator value on the basis of the inference set. The inference set comprises the plurality of J first error- mitigated target estimator values E^Aj) inferred in the inference step for each of the auxiliary quantum circuits Aj.

[0077] The final target estimator value is an error-mitigated target estimator value. Due to the special structure of the noise-reduced target estimator values with the first and second contributions, the final target estimator value inferred according to the method of the present invention may be closer to the expectation value of the observable in the absence of any noise and may further be more robust against an inaccuracy in noise scaling and / or a choice of a spacing of the noise strength values.

[0078] In one embodiment of the method according to the first aspect of the present invention, the routine may further comprise for each auxiliary quantum circuit:

[0079] calculating a value of a normalized measure of dispersion for the associated noise- reduced target estimator values with the optimal control parameter value, the value of the normalized measure of dispersion being indicative of a ratio of a value of a first measure of dispersion for the noise-reduced target estimator values with the optimal control parameter P-00183WO 05122025 value associated with the auxiliary quantum circuit and a value of the first measure of dispersion for the target estimator values;

[0080] and wherein the inference set comprises for each first error-mitigated target estimator value the associated value of the normalized measure of dispersion.

[0081] I,e., according to the embodiment, the final target estimator value is inferred on the basis of the inference set which comprises the plurality of first error-mitigated target estimator values and the associated values of the normalized measure of dispersion.

[0082] In one expedient example of the embodiment, the first measure of dispersion may be a function of or may be equal to the variance, the standard deviation or the mean absolute difference, but the invention is not limited to this. The normalized measure of dispersion may be a function of or may be equal to the ratio of the value of the first measure of dispersion for the noise-reduced target estimator values with the optimal control parameter value and the value of the first measure of dispersion for the target estimator values. In one example, the normalized measure of dispersion may be of the following form:

[0083] Differed

[0084] Av(A) “ - - p -

[0085]

[0086] Here, Drdenotes the first measure of dispersion, ^Ereddenotes the

[0087]

[0088] set of the M noise-reduced target estimator values (m - 1,..., M) for the optimal control parameter value njiOp, and {E denotes a set of the plurality of M target estimator

[0089]

[0090] values

[0091]

[0092] M) of the target set.

[0093] In one example of the embodiment, inferring the final error-mitigated target estimator value may be on the basis of the first error-mitigated target estimator value with the smallest value of the normalized measure of dispersion among the plurality of first error-mitigated target estimator values. Thereby, the final error-mitigated target estimator value may be doser to the noiseless expectation value of the observable and may further be more robust against an inaccuracy in noise scaling and / or a choice of the spacing of the noise strength values. In one example, the final error-mitigated target estimator value may be equal to the first error- mitigated target estimator value with the smallest value of the normalized measure of dispersion.

[0094] P-00183WO 05122025 However, the embodiment is not limited to this, and in another example of the embodiment, inferring the final error-mitigated target estimator value comprises selecting a set of auxiliary quantum circuits among the plurality of auxiliary quantum circuits to thereby define a data set comprising for each auxiliary quantum circuit of the set a tuple of the derived first error-mitigated target estimator value and the associated value of the normalized measure of dispersion, defining a fitting model for the data set, fitting the fitting model to the data set and extrapolating to the limit of zero for the value of the normalized measure of dispersion to thereby infer the final error-mitigated target estimator value. In one example, selecting the set of auxiliary quantum circuits may comprise selecting the plurality of J auxiliary quantum circuits. In another example, the set of selected auxiliary quantum circuits may comprise a proper subset of the plurality of J auxiliary quantum circuits. Further examples how to select the set of auxiliary quantum circuits are presented below.

[0095] The data set comprises the tuples (Ei(Aj), DN (Aj)), wherein j e J, with / , being an index set of the indices j of the selected auxiliary quantum circuits, Ei(Aj) is the first error-mitigated target estimator value associated with the auxiliary quantum circuit Aj and DN(AJ) is the value of the normalized measure of dispersion associated with the first error-mitigated target estimator value Ei(Aj).

[0096] The tuples of the first error-mitigated target estimator value and the associated value of the normalized measure of dispersion (Ei(Aj), DN(A,)) are used to infer the final error-mitigated target estimator value at zero normalized measure of dispersion by extrapolation. Thereby, the bias error of the final error-mitigated target estimator value may be reduced compared to methods of the prior art. In this way, the accuracy of the final error-mitigated target estimator value may be improved compared to the accuracy of the first error-mitigated target estimator value and may also be improved compared to the accuracy of the first error-mitigated target estimator with the smallest value of the normalized measure of dispersion among the plurality of first error-mitigated target estimator values. This embodiment of the method has some robustness against the inaccuracy of noise scaling, the choice of the noise level spacing, the choice of the auxiliary quantum circuit, the circuit depth of the target and auxiliary circuits, the readout error and the coherent error.

[0097] In one example, the final error-mitigated target estimator value may be equal to the extrapolated first error-mitigated target estimator value at the value of zero for the normalized measure of dispersion.

[0098] P-00183WO 05122025 In one example of the embodiment, selecting the set of auxiliary quantum circuits may comprise selecting only those auxiliary quantum circuits for which the associated value of the normalized measure of dispersion is below a predetermined threshold value. This may be understood as a filtering process. In one example, the threshold value may be 3 or smaller, preferably 2.5 or smaller, more preferably 2 or smaller, and even mor preferably 1.5 or smaller.

[0099] In one expedient embodiment of the method according to the first aspect of the present invention, the noise-reduced target estimator value may be derived in the noise-reduction step such that in the zero-noise limit the noise-reduced target estimator value is equal to the target estimator value, and wherein inferring the first target estimator value in the inference step may comprise estimating the zero-noise limit of the noise-reduced target estimator value with the optimal control parameter value. In particular, this equality holds independent of the value of the control parameter. Thereby, the first error-mitigated target estimator value may be determined as the zero-noise limit of the noise-reduced target estimator value in the inference step. However, the invention is not limited to this, and the noise-reduced target estimator value may be derived such that in the zero-noise limit the noise-reduced target estimator value is proportional to the target estimator value or a value of an invertible function applied to the target estimator value in certain examples. The embodiment allows for a particularly simple way to infer the first target estimator value on the basis of the noise-reduced target estimator values.

[0100] In another embodiment of the method according to the present invention, the method may be such that for each auxiliary quantum circuit deriving the first contribution of the noise-reduced target estimator value for each target noise strength value comprises assigning one auxiliary noise strength value to the target noise strength value and dividing the target estimator value associated with the target noise strength value by the auxiliary estimator value associated with the auxiliary quantum circuit and the assigned auxiliary noise strength value and by multiplying the result with the noise-free auxiliary value associated with the auxiliary quantum circuit.

[0101] In particular, in the embodiment the assignment of the auxiliary noise strength values may be one-to-one, i.e., the number of target and auxiliary noise strength values is identical, and there is a bijective function such that

[0102]

[0103] = G(A® ). In one expedient example, A^ = xA® for some positive x>0. In particular, x=1. That is, the first contribution may be of the form

[0104] P1 E(0,Aj).

[0105]

[0106] P-00183WO 05122025 The first contribution Px, ) is expected to have a reduced

[0107]

[0108] noise-dependency compared to the target estimator values

[0109]

[0110] in particular when the target and auxiliary estimator values have a similar noise-dependency.

[0111] In another embodiment of the method according to the first aspect of the present invention, the control parameter-dependent model for the remaining noise-dependency of the first contribution may comprise for each auxiliary quantum circuit a term which is a product of the control parameter and a noise-cancelling function in the noise strength which depends on the auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit and has a noise-dependency which is reduced compared to the noisedependency of the auxiliary estimator values associated with the auxiliary quantum circuit.

[0112] The parameter-dependent model may be expressed as

[0113] njfj {E, E(0, A;)),

[0114]

[0115] wherein fj is the noise-cancelling function and rtj is the control parameter. The noise cancelling function fj may be the same noise cancelling function for each auxiliary quantum circuit, but the invention is not limited to this. In one expedient example, the control parameter is real-valued and / or the noise-cancelling function is real-valued. The control parameterdependent model of the embodiment is linear in the control parameter

[0116]

[0117] In one example, the value of the noise-cancelling function fj for each target noise strength value may depend only on the auxiliary estimator value for the assigned auxiliary noise strength value

[0118]

[0119] and the noise-free auxiliary estimator value, i.e.,

[0120] P2( A®, • E(0, A,), n;) =n, / ;(E(G(A®), Aj). E(0, A,)).

[0121]

[0122] In one expedient example, the noise-canceling function may be of the form

[0123] f (E (G(^). Aj). E(0, A;)) = log ■

[0124]

[0125] In this case, the second contribution may be of the following form:

[0126] P2U®, E(G(l<;>), AJ), E(0, A;),n,.) = n, log (

[0127]

[0128] jfAj)

[0129] P-00183WO 05122025 In one expedient example where A® = (1®) =

[0130]

[0131] $w'^h x>0, the noise-reduced target estimator value may be of the form

[0132] (C + n, log (.

[0133]

[0134] In particular, x=1 in one example.

[0135] In general, one expects that the auxiliary estimator values decay exponentially with the noise strength value. Thereby, the logarithmic function of the model has a noise-dependency which is reduced compared to the noise-dependency of the auxiliary estimator values.

[0136] In one further embodiment of the method according to the first aspect of the present invention, the minimization criterion may be fulfilled for at least one auxiliary quantum circuit in the control parameter step when a zero-noise limit of a K-th Taylor polynomial of the noise- reduced target estimator value in the noise strength about an expansion value selected among the plurality of target noise strength values, wherein derivatives are replaced by difference quotients which are expressed in terms of the parameter-dependent noise-reduced target estimator values, is equal to the noise-reduced target estimator value associated with the expansion value, wherein,

[0137] inferring the first error-mitigated target estimator value in the inference step is on the basis of the noise-reduced target estimator value associated with the expansion value and obtained for the optimal control parameter value, and wherein

[0138] preferably the target noise strength values increase with the target noise level, and the expansion value is the smallest target noise strength value of the plurality of target noise strength values.

[0139] Preferably, the minimization criterion is the same for all auxiliary quantum circuits, but the invention is not limited to this. The expansion value may be the same for all auxiliary quantum circuits in one expedient example, but the invention is not limited to this.

[0140] In one example, the first-error mitigated target estimator value may be determined as the noise-reduced target estimator value for the expansion value and for the optimal control parameter value.

[0141] P-00183WO 05122025 For the auxiliary quantum circuit Aj, the K-th Taylor polynomial

[0142]

[0143] of the noise- reduced target estimator value in the noise strength about an expansion value 2® selected among the plurality of target noise strength values is of the form

[0144] KCl - 2(t)

[0145] 4’ j, +

[0146]

[0147] k=l

[0148] wherein E^d(Aj, A^eis a k-th order difference quotient which approximates the k- th derivative | co of the noise-reduced target estimator value at the expansion

[0149]

[0150] az z-zmeJ.

[0151] value 2^. A k-th order difference quotient <?[k]which approximates the k-th derivative

[0152]

[0153] I, of a function 0(2) in the noise strength at the value 2®;may be represented as azK^=VeJ-meiJ

[0154] )> wherein are real coefficients for the auxiliary quantum

[0155]

[0156] circuit Aj, and k < M - 1, wherein M is the total number of target noise strength values used in the method. I.e., given M target noise strength values, difference quotients up to order M-1 can be calculated. Note that the coefficients

[0157]

[0158] are different for different auxiliary quantum circuits Aj only and only if the noise-level scaling and the expansion values are different for the different auxiliary quantum circuits. If, however, the noise-level spacing and the expansion values are the same for all auxiliary quantum circuits, then the coefficients

[0159]

[0160] notdepend o

[0161]

[0162] n j, i.e., a"4 = a^m.

[0163] The number of non-zero coefficients

[0164]

[0165] may be related to the accuracy of the approximation for a given order k of the derivative. Note that the accuracy of the approximation also depends on the number M of noise scales. The difference quotient may be a forward, a backward or a central difference quotient in one example.

[0166] In one expedient example, the expansion value is the smallest target noise strength value associated with the smallest target noise level, in particular 2® j = 1. However, the invention is not limited to this, and the expansion value may be a target noise strength value which is different from the smallest target noise strength value in certain examples.

[0167] In one example of the embodiment, the method may comprise calculating the zeronoise limit of the K-th Taylor polynomial, i.e., TjM(0), and the optimal control parameter value

[0168] P-00183WO 05122025 njiOpmay be determined such that r](0) = Ered(Aj, A^ej,njiOpf I.e., the sum of all higher order contributions of the K-th Taylor polynomial vanishes for the optimal control parameter

[0169] value n

[0170]

[0171] .jiOp, Ef=i ^Ld=0-

[0172] In certain instances, the K-th Taylor polynomial needs not to be explicitly calculated, though. One particular example is the case when the noise-reduced target estimator values are a sum of the first and second contributions which are functions of the target and auxiliary estimator values for the target noise strength values and the assigned auxiliary noise strength values, respectively, and the model is linear in the control parameter, i.e., Ered=

[0173]

[0174] Pl (E^, T), E(G + njfjCA^MG (<}), Aj), E(0, Aj))). In one particular example,

[0175] fj ^A^, E(G If the k-th difference quotients of the first

[0176] contribution (E (A^, T), E(G and the noise-canceling function fj at the expansion

[0177]

[0178] value are expressed as p}k]J, T), E(G (2^7.)M;)) =

[0179] (E(AP, T), E(G (A®),^)) and fjlk](A^eJ, E(G «y), Ajf E(p, Aj^ =

[0180]

[0181] (A^, Aj), E(0, Aj))), respectively, the optimal control parameter value may

[0182]

[0183] be calculated according to

[0184] g=1ViKiPi n Me (4°) >4-))

[0185] n

[0186]

[0187] j,op~

[0188] ( jp) \k

[0189] wherein v}= £fc=i"

[0190]

[0191] — '^~aki for the auxiliary quantum circuit Aj. I. e, the optimal control parameter for the auxiliary quantum circuit Aj may be calculated from the values of the first contribution P1(E(A[t\T), E(G (A^ ^j)^, the values of the noise-cancelling function

[0192]

[0193] fj (A^, E (G(A^), Aj^, E(0, Aj))^ and the coefficients a“ of the k-th difference quotients.

[0194]

[0195] In one example, the noise-free auxiliary values may be provided as an input of the method. However, the method is not limited to this, and in one further embodiment, the method may comprise determining the noise-free auxiliary value for at least one auxiliary quantum circuit, wherein the determining comprises a classical simulation of the auxiliary quantum circuit at zero noise on a classical computer. Thereby, the noise-free auxiliary value may be determined as an exact value or an estimator value for the expectation value of the observable. P-00183WO 05122025 In one expedient example, the method comprises determining the noise-free auxiliary value for each auxiliary quantum circuit by classical simulation.

[0196] The method according to the present invention is configured to be applied to any target and auxiliary sets that are obtained for an execution of the target and auxiliary circuits, respectively, for each of the plurality of target and auxiliary noise strength values, respectively, each target (auxiliary) noise strength value being associated with one of the target (auxiliary) noise levels. In one embodiment of the method, the target and auxiliary quantum circuits may be error-corrected quantum circuits according to a quantum error correction code, and the execution of the target and auxiliary quantum circuits at the plurality of different noise levels may comprise their execution for a plurality of different code distances of the quantum error correction code. This method is called code distance scaling within this specification. Code distance scaling is disclosed, e.g., in M. Wahl et.al, “Zero noise extrapolation on logical qubits by scaling the error correction code distance”, arxiv:2304.14985. In this way, the method according to the present invention may be applied to logically encoded qubits.

[0197] In a further embodiment of the present invention, the execution of the target and auxiliary quantum circuits at the plurality of different noise levels may comprise one of the following such as unitary folding on physical or logical qubits, local gate folding, noise amplification via pulse level control or probabilistic noise-amplification. Unitary folding and local gate folding are explained in more detail, for example in T. Giurgica-Tiron et.al, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), (IEEE, 2020). Noise amplification via pulse-level control comprises changing a duration of a control pulse used for an implementation of a quantum gate, see e.g. K. Temme et.al, Phys. Rev. Lett. 119, 180509 (2017). Probabilistic noise amplification is disclosed, for example in Y. Kim et.al, Nature 618, 500 (2023).

[0198] The method according to the present invention is applicable to any target and auxiliary sets, irrespective of how these sets are obtained. That is, the target and auxiliary sets may be seen as an input of the method in one example. However, the method is not limited to this, and in one further embodiment of the method according to the present invention, the method may further comprise executing the target quantum circuit and / or the plurality of auxiliary quantum circuits by the quantum processor at each of the plurality of target and / or auxiliary noise levels to thereby obtain the target measurement data and / or the plurality of auxiliary measurement data, respectively, and evaluating the target and / or auxiliary measurement data by the classical computer to thereby determine the target and / or auxiliary estimator values, respectively.

[0199] P-00183WO 05122025 In one expedient example of the embodiment, the method may comprise assigning an auxiliary noise strength value to each target noise strength value, and executing the target quantum circuit and the plurality of auxiliary quantum circuits may be such that for each target noise strength value and the assigned auxiliary noise strength value the target quantum circuit and all auxiliary quantum circuits of the plurality of auxiliary quantum circuits are executed consecutively at a target noise level associated with the target noise strength value and at an auxiliary noise level associated with the assigned auxiliary noise strength value, respectively. Le., for each target noise strength value A® there is one assigned auxiliary noise strength value A^ = G ( ® ) assigned one-to-one and the target circuit T and the J auxiliary quantum circuits Aj, j = 1,.... J are executed consecutively at a target noise level associated with the target noise strength value A^ and at an auxiliary noise level associated with the assigned auxiliary noise strength value A^ = G (A®), respectively. Quantum processors are prone to temporal instabilities including parameter drift. Thereby, properties of the qubits, the applied quantum gates, the quantum measurement, the prepared initial state, etc. which are relevant for an execution of a quantum circuit may be time-dependent. This may negatively affect any quantum error mitigation method. The effect of the parameter drift may be reduced by executing the target and auxiliary quantum circuits consecutively for each target noise strength value and the assigned auxiliary noise strength value according to the embodiment, preferably followed by a recalibration of the quantum processor before the target and auxiliary quantum circuits are executed at another value of the target noise strength and the assigned auxiliary noise strength value, respectively.

[0200] According to a second aspect of the present invention, there is provided a computer program product comprising instructions which, when the computer program product is executed by a classical computer, cause the classical computer to carry out the method according to any one of the above embodiments.

[0201] According to a third aspect of the present invention, there is provided a data carrier having stored thereon the computer program product according to the second aspect of the present invention.

[0202] In particular, the data carrier may be a non-transitory data carrier in one example.

[0203] According to a fourth aspect of the present invention, there is provided a computing system comprising a classical computer and a quantum processor, wherein

[0204] P-00183WO 05122025 the quantum processor is configured to execute the target and auxiliary quantum circuits at each of the plurality of noise levels to thereby obtain the target and auxiliary measurement data, respectively,

[0205] wherein the classical computer is configured to evaluate the target and auxiliary measurement data to thereby determine the target and auxiliary estimator values, respectively, and

[0206] wherein the classical computer is further configured to execute the computer program product according to the second aspect of the present invention.

[0207] Everything that was said above in relation to the method according to the first aspect of the present invention and the computer program product according to the second aspect of the present invention also applies to the computing system according to the fourth aspect of the present invention. The classical computer and the quantum processor may be located at the same location in one example. In another example, the quantum processor and the classical computer may be located at different locations. In one example, at least one of the quantum processor and the classical computer may be a cloud processor / computer.

[0208] In the following, the invention is described in more detail by way of example with reference to the Figures, in which

[0209] Fig. 1 is a schematic representation of a computing system according to the fourth aspect of the present invention;

[0210] Fig. 2 depicts a flowchart of method steps of a first embodiment of the method according to the first aspect of the present invention;

[0211] Fig. 3 depicts a flowchart of method steps of a second embodiment of the method according to the first aspect of the present invention;

[0212] Fig. 3a depicts a flowchart of further method steps of a first example of the second embodiment of the method according to the first aspect of the present invention; Fig. 3b depicts a flowchart of further method steps of a second example of the second embodiment of the method according to the first aspect of the present invention; Fig. 4a schematically depicts a system of ten qubits of a quantum processor of the computing system of the fourth aspect of the present invention and their connectivity; Fig. 4b depicts results of the method according to the second example of the second embodiment, wherein the target and auxiliary sets are obtained by a classical simulation of the execution of target and auxiliary quantum circuits on the qubits of the quantum processor of Fig. 4a;

[0213] P-00183WO 05122025 Fig. 4c depicts further results of the method according to the second example of the second embodiment, wherein the target and auxiliary sets are obtained by the classical simulation of the execution of the target and auxiliary quantum circuits on the qubits of the quantum processor of Fig. 4a.

[0214] Fig. 1 depicts a schematic representation of a computing system 20 according to the fourth aspect of the present invention. The computing system 20 comprises a quantum processor 1 and a classical computer 10. Furthermore, the computing system 20 comprises an interface 50 interfacing the quantum processor 1 and the classical computer 10.

[0215] The classical computer 10 comprises a classical processor 11, memory means 12, and an input / output unit 13. A computer program is stored on the memory means 12 of the classical computer and the classical processor 11 is configured to execute the computer program stored on the memory means 12. The operation of the classical computer is based on operations on classical bits 0 and 1.

[0216] The classical computer 10 is configured to receive a user input via the input / output unit 13 in the form of classical data and to provide and receive data and / or instructions via the interface 50 to and from the quantum processor 1.

[0217] When the computer program is executed by the classical processor 11, the classical computer 10 prompts the user to provide an input required for the execution of the computer program stored on the memory means 12. The input comprises data specifying a target quantum circuit defined on N ≤ Nmaxqubits and optionally at least one feature of a set of features comprising a noise scaling method, a plurality of target noise strength values, a method for deriving a plurality of auxiliary quantum circuits from the target quantum circuit, a plurality of auxiliary noise strength values, a number S of shots for the execution of the quantum circuits, and an observable. Those features which may not be selected by the user may be preset in the computer program stored on the memory means 12.

[0218] The instructions specifying the target quantum circuit comprise instructions specifying an initial target state of the N qubits, a target gate sequence, and a target measurement. The target quantum circuit is a quantum circuit of interest and configured to solve a computational problem of interest. An example of such a problem will be presented below.

[0219] The noise scaling method may comprise at least one of unitary folding on physical or logical qubits, local gate folding, noise amplification via pulse level control, probabilistic noise

[0220] P-00183WO 05122025 amplification or code distance scaling. Recall that within this specification, code distance scaling refers to a scaling method for the case that the target and auxiliary quantum circuits are error-corrected quantum circuits according to a quantum error correction code, and the execution of the target and auxiliary quantum circuits at the plurality of different target and auxiliary noise levels comprises their execution for a plurality of different code distances of the quantum error correction code, respectively (see e.g. arXiv:2304.14985 cited above).

[0221] Each noise scale factor is associated with a noise level for the execution of the quantum circuit. Within the field of quantum error mitigation, the hardware base noise is in general associated with a noise strength value 1 = 1. All other noise strength values are larger than 1. For ease of explanation, the target and auxiliary noise strength values are equal in number and values in the following example, i.e., λ(t)m= λ(a)m= λmfor all m = 1,..., M. For simplicity,mis denoted as noise strength value in the following. The plurality of noise strength values comprises at least three noise strength values, i.e., 1 = Ai < li- <... < AM, M > 3. In particular, M=3, 4, 5 or more in one example. The spacing between two consecutive noise strength values, Am+i - Am, may be the same or may be different from each other.

[0222] In one example, the computer program may comprise instruction which, when the computer program is executed by the classical processor 11, cause the classical processor 11 to derive a plurality of J ≥ 2 auxiliary quantum circuits Aj, j=1, J from the target quantum circuit. For example, J=50, 100, 150. The auxiliary quantum circuits are derived such that they are all different from each other and different from the target quantum circuit. Each auxiliary quantum circuit comprises instructions specifying an auxiliary initial state, an auxiliary gate sequence, and an auxiliary measurement In particular, the auxiliary initial state may be identical to the initial target state, and the auxiliary measurement may be identical to the target measurement for all auxiliary quantum circuits. In the example, each auxiliary gate sequence is derived from the target gate sequence by replacing a large fraction of the non-Clifford gates, in particular around 80 % of the non-Clifford gates or all non-Clifford gates, by a Clifford gate and by keeping the rest of the non-Clifford gates. In this way, the auxiliary gate sequence has a structure which is similar to the structure of the target gate sequence and which may be simulable on a classical computer. The non-Clifford gates which are replaced may be selected randomly in one example. In one example, the angle of each Clifford gate is selected to minimize the distance to the original non-Clifford angle of the replaced non-Clifford gate. However, the invention is not limited to this, and other ways to derive the auxiliary gate sequence from the target gate sequence may be envisioned. Furthermore, when the computer program is executed by the classical computer, instructions specifying the target and auxiliary quantum gate sequences and their implementation for the plurality of noise strength values

[0223] P-00183WO 05122025 are created. These instructions are then provided to the quantum processor 1 by the interface 50.

[0224] The quantum processor 1 comprises a register 2 of Nmax≥ 2 qubits, state preparation means 3 configured to prepare a predetermined initial state, means 4 for applying quantum gates to the qubits of the register 2, and measurement means 5 configured to apply a quantum measurement to the qubits of the register 2. The qubits 2 of the quantum processor 1 may be any type of qubits, including, but not limited to superconducting qubits. Furthermore, the quantum processor 1 comprises a classical processor / controller 6 configured to receive instructions from the classical computer 10 / interface 50. These instructions comprise instructions for the execution of the target quantum circuit and each auxiliary quantum circuit for each of the plurality of different noise strength values.

[0225] The quantum processor 1 is configured to execute, by control of the controller 6, the target quantum circuit and each auxiliary quantum circuit for the plurality of different noise strength values and the predetermined number of shots. The quantum measurement implemented by the measurement means 5 results for each value Amof the noise strength and each shot s of the S shots in associated target and auxiliary measurement outcomes, respectively. Thereby, target and auxiliary measurement data may be created which comprises for each noise strength value the associated target and auxiliary measurement outcomes, respectively. The target and auxiliary measurement data is provided via the interface 50 to the classical computer 10. The computer program is executed by the classical processor to thereby evaluate the target and auxiliary measurement data and to determine for each noise strength value Amthe associated target estimator value, E (Am, T), and the auxiliary estimator values associated with the respective auxiliary quantum circuit Aj, E (Am, Aj), for the expectation value of the observable O. (As mentioned above, the target and auxiliary noise strength values are equal in number and values in the example.) The quantum measurement is such that the target and auxiliary estimator values of the observable O may be obtained from the target and auxiliary measurement data, respectively. Furthermore, when the computer program is executed by the classical processor 11, the classical processor 11 simulates the execution of each auxiliary quantum circuit Aj at zero noise to thereby determine the associated noise-free auxiliary value of the expectation value of the observable O, E (0, Aj), or its estimator value.

[0226] The computer program further comprises instructions, which, when the computer program is executed by the classical processor 11, cause the classical computer to implement one of the embodiments and examples of the method according to the first aspect of the present invention explained in more detail below with reference to Figs. 2 to 6.

[0227] P-00183WO 05122025 Fig. 2 depicts a flowchart of method steps of a first embodiment of the method according to the first aspect of the present invention. At step S1, a target set comprising for each of the noise strength values the associated target estimator value and a plurality of J ≥ 2 auxiliary sets, each comprising for each of the noise strength value the associated auxiliary estimator value for the respective auxiliary quantum circuit, and the associated J noise-free auxiliary values are received. These sets may, e.g., be generated by the computing system 20 according to the second aspect of the present invention described above with reference to Fig. 1.

[0228] The method continues with a first variant of a processing step S2. There, a routine is performed for each auxiliary quantum circuit Aj. The routine comprises the following steps:

[0229] At step S2a, a noise reduction step is implemented. In the noise reduction step, a control parameter-dependent noise-reduced target estimator value is derived for each noise strength value Am. The noise-reduced target estimator value consists of first and second contributions and is of the following form:

[0230] E

[0231]

[0232] red^npAj} = P1(E(A.rn, T'), E(Ain, Aj) ) + P2«Am, AJ), <0, Aj^rij).

[0233] The first contribution is of the following form:

[0234] P, E(0M;).

[0235]

[0236] The second contribution is of the following form:

[0237] P2(El^Aj^ElO. A^nj) = n}log

[0238]

[0239] The noise-reduced target estimator value is linear in the real control parameter rij. In the zero-noise limit, the noise-reduced target estimator value is equal to the target estimator value irrespective of the value of the control parameter, and irrespective of the auxiliary quantum circuit, Ered0,nj, Aj) = E(0, T).

[0240] The method proceeds with a control parameter step S2b. In the control parameter step, an optimal control parameter value n.japis determined. To this end, an expansion value

[0241]

[0242] is selected among the plurality of noise strength values. Then, the optimal control parameter value njtOpis determined such that the zero-noise limit of a K-th Taylor polynomial of the noise-

[0243] P-00183WO 05122025 reduced target estimator value in the noise strength about the expansion value, wherein derivatives are replaced by difference quotients which are expressed in terms of the noise- reduced target estimator value is equal to the noise-reduced target estimator value associated with the expansion value. I.e., at the optimal control parameter value nj opthe sum of all higher order contributions of the K-th Taylor polynomial vanish in the zero-noise limit. In particular, the expansion value

[0244]

[0245] is the smallest noise strength value of the plurality of noise strength values, i.e., = 1. As has been explained above in the general part of the description, an explicit calculation of the K-th Taylor polynomial is not necessary. Rather, the optimal control parameter value may be calculated according to

[0246] _ n.g(4 AjS)

[0247] j,opyf

[0248]

[0249] 2,1=1« >°glEn.A.) I

[0250] wherein

[0251] VKj _vK(~l)fcMj

[0252]

[0253] — Lk=!fc!akl ’

[0254] and the coefficients

[0255]

[0256] are coefficients for calculating the k-th different quotients of the derivatives of the first contribution and the noise-cancelling function, Plk\E(XmeJ, T), E(AmgJ, Aj)) = ^=1^ljP1(E(AbT), E^, Aj)) and flk]^^mej, Aj,), E(0, Aj))') = ^a^fCE^A^E^Aj)) with f (E(XbAj), E(Q, Ay)) =

[0257]

[0258] log 1. See the general part of the description for more detail.

[0259] The method proceeds with the (first) inference step at S2c. There, the first error mitigated target estimator value E^Aj) is determined as the noise-reduced target estimator value associated with the expansion value

[0260]

[0261] 2me>y= = 1 and obtained for the optimal control parameter value ny>op.

[0262]

[0263] =Ered(Aj,

[0264] After the routine is executed for each auxiliary quantum circuit Aj, j=1,..., J, the method proceeds with the final inference step S3. There, the final error-mitigated target estimator value is inferred from the plurality of J first error-mitigated target estimator values.

[0265] P-00183WO 05122025 Fig. 3 depicts a flowchart of method steps of a second embodiment of the method according to the first aspect of the present invention. The method according to the second embodiment starts with the data step S1 explained with reference to Fig. 1. Then, the method proceeds with the processing step 2’ (see Fig. 2) which is a modification of the processing step of Fig. 2. In particular, the processing step S2’ comprises the steps 2a, 2b and 2c of the processing step S2 of Fig. 2 and the additional method steps 2d. The order of the steps 2c and 2d may be interchanged in one example.

[0266] At step S2d, a value DN(A7) of a normalized measure of dispersion for the noise-reduced target estimator values is calculated. The value of the normalized measure of dispersion DN is defined as the quotient of a value of a first measure of dispersion Di for the noise-reduced target estimator values for the optimal value of the control parameter and a value of the first measure of dispersion for the target estimator values, that is

[0267] n ri \ Gm>ny,op4j)}m=1)

[0268]

[0269] °K&4') =

[0270] Di is the first measure of dispersion, and may be, for example, the variance, the standard deviation or the mean absolute difference. {Ered{xm,nj^, A^^^ denotes the set of the

[0271]

[0272] noise-reduced target estimator values for the plurality of different noise strength values and the optimal control parameter value. {E (mT)}^=1denotes the set of the target estimator values for the plurality of different noise strength values. Thereby, a plurality of first error-mitigated target estimator values, each associated to one of the auxiliary quantum circuits, and associated values of the normalized measure of dispersion are obtained.

[0273] Fig. 3a depicts a flow chart comprising a final inference step S31 of a first example of the second embodiment of the method according to the present invention. The final inference step S31 is after the processing step S2’.

[0274] At S31, the final error-mitigated target estimator value is determined as the first error-mitigated target estimator value with the smallest value of the normalized measure of dispersion DNamong the plurality of first error-mitigated target estimator values.

[0275] Alternatively, in a second example of the second embodiment of the method according to the present invention, the final error-mitigated target estimator value is determined according to the method steps S32 and S33 shown in Fig. 3b.

[0276] P-00183WO 05122025 First, a plurality of auxiliary quantum circuits / first error-mitigated target estimator values E^Aj), j e J is selected among the plurality of first error-mitigated target estimator values. Here, / is an index set of indices of the selected auxiliary quantum circuits. In the example, those first error-mitigated target estimator values are selected for which the value of the normalized measure of dispersion is equal to or smaller than a predetermined threshold value. In the example specified below, the threshold value is 2. However, other threshold values are also be possible in other examples. Next, a data set is constructed which comprises tuples of the selected first error-mitigated target estimator values and the associated value of the normalized measure of dispersion.

[0277] The method proceeds to step S33. There, a linear fitting model is defined, the fitting model is fitted to the data set and extrapolated to the limit of zero for the value of the normalized measure of dispersion. However, the invention is not limited to a linear fit. The final error- mitigated target estimator value is determined as the extrapolated first error-mitigated target estimator value for the value of zero for the normalized measure of dispersion.

[0278] In the following, results obtained with the method according to the first aspect of the present invention for mitigating the error caused by noise in an execution of a specific target quantum circuit will be presented. The target quantum circuit T is a QAOA (Quantum Approximate Optimization Algorithm)-type quantum circuit configured to determine the ground state of a transverse Ising model Hamiltonian H =

[0279]

[0280] ~ wherein

[0281]

[0282] is the Pauli X-matrix acting on the qubit j, o is the Pauli Z-matrix acting on qubit j, and

[0283]

[0284] az ^°z ’s acoupling term between connected qubits j and j' of the quantum processor. All results are obtained for the case g=2. The target quantum circuit comprises a target gate sequence comprising single-qubit rotations around the X, Y and Z axis and controlled-Z operations and a measurement in the Z-basis. The target measurement results in target measurement data which allows to calculate an estimator value of an expectation value of the ground state energy of the transverse Ising model, which is the observable in the present example. Then, 51 auxiliary quantum circuits are derived from the target quantum circuit. Each auxiliary quantum circuit comprises the same initial state and the same quantum measurement as the target quantum circuit. For 50 of the auxiliary quantum circuits, the auxiliary gate sequence is derived from the target gate sequence of the target quantum circuit by selecting about 80 % of the non-Clifford gates and by replacing each of the selected non-Clifford gates by the closest Clifford gate as explained above. One of the auxiliary quantum circuits is derived

[0285] P-00183WO 05122025 by replacing each non-Clifford gate with closest Clifford gate (i.e., the Clifford gate for which the distance between the Clifford angle and the non-Clifford angle is minimal).

[0286] Fig. 4a schematically represents a register of 10 qubits 100 of a quantum processor 1

[0287]

[0288] Fig. 4b depicts data sets of tuples of the first error-mitigated target estimator values, normalized by the exact target estimator value, E1(

[0289]

[0290] Aj) / < O>, and the associated value DN(

[0291]

[0292] Aj)

[0293]

[0294] h=1, h=2 (i.e., noise strength values [1, 1+h, 1+2h]) by applying the method according to the second example of the second embodiment and using S = 5-104shots for the execution of

[0295]

[0296] are near-Clifford circuits. The x-axis indicates the value of the normalized measure of dispersion and the y-axis the normalized first error-mitigated target estimator value E1 / <0>

[0297]

[0298] Clifford circuit. The star 43 indicates the final error-mitigated target estimator value obtained

[0299]

[0300] error-mitigated target estimator value E1 / <0>, which is 1 The results shown in Fig. 4c are obtained in a similar way as the results shown in Fig.

[0301] 4b. The only difference is the number of shots. To be precise, only 2.5- 103shots were used for the execution of each quantum circuit and a given noise scale factor. Again, the dots 41 indicate the results obtained for the 50 different near-Clifford circuits and the triangle 42 indicates the result for the full Clifford circuit. The star 45 indicates the final error-mitigated target estimator obtained by a linear fit to the data set comprising the 51 error-mitigated target estimator values and the associated value of the normalized measures of dispersion and by an extrapolation to the value of zero for the normalized measure of dispersion. The star 46 indicates the final error-mitigated target estimator value obtained by a linear fit to a data set which comprises only those first error-mitigated target estimator values and the associated values of the normalized measure of dispersion for which the value of the normalized measure of dispersion is smaller than 2. The error bars are obtained via bootstrapping.

[0302] P-00183WO 05122025

Claims

PATENT CLAIMS1. A method for mitigating errors caused by noise in a target quantum circuit executed by a quantum processor on the basis of a target set associated with the target quantum circuit, a plurality of auxiliary sets, each being associated with a distinct auxiliary quantum circuit of a plurality of auxiliary quantum circuits each being different from the target quantum circuit, and for each of the auxiliary quantum circuits an associated noise-free auxiliary value, said target set comprising for each of a plurality of target noise strength values an associated target estimator value for an expectation value of an observable, the target estimator values being determined by evaluation of target measurement data obtained by an execution of the target quantum circuit by the quantum processor at a plurality of different target noise levels, each target noise level being associated with one of the target noise strength values, each auxiliary set comprising for each of a plurality of auxiliary noise strength values an associated auxiliary estimator value for an expectation value of the observable, the auxiliary estimator values being determined by evaluation of associated auxiliary measurement data obtained by an execution of the auxiliary quantum circuit associated with the auxiliary set by the quantum processor at a plurality of different auxiliary noise levels, each auxiliary noise level being associated with one of the auxiliary noise strength values and each noise-free auxiliary value being an expectation value of the observable for a fictitious noise-free execution of the associated auxiliary quantum circuit or an estimator value thereof, said method comprising:for each of the auxiliary quantum circuits, executing a routine for inferring a first error-mitigated target estimator value associated with the respective auxiliary quantum circuit, the routine comprising:a noise-reduction step of deriving for each target noise strength value a noise- reduced target estimator value associated with the auxiliary quantum circuit, said noise- reduced target estimator value comprising first and second contributions, wherein the first contribution is derived by reducing a noise-dependency of the target estimator values on the basis of the auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit, and the second contribution is configured to further reduce a remaining noise-dependency of the first contribution and is derived by use of a control parameter-dependent model for the remaining noise dependency on the basis of the auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit;P-00183WO 05122025a control parameter step of determining an optimal value of the control parameter such that the noise-dependency of the noise-reduced target estimator values fulfills a minimization criterion;an inference step of inferring the first error-mitigated target estimator value associated with the auxiliary quantum circuit from the noise-reduced target estimator values with the optimal control parameter value associated with the auxiliary quantum circuit;said method further comprising:inferring a final error-mitigated target estimator value on the basis of an inference set which comprises the plurality of first error mitigated target estimator values.

2. Method according to claim 1, wherein the routine further comprises for each auxiliary quantum circuit:calculating a value of a normalized measure of dispersion for the associated noise-reduced target estimator values with the optimal control parameter value, the value of the normalized measure of dispersion being indicative of a ratio of a value of a first measure of dispersion for the noise-reduced target estimator values with the optimal control parameter value associated with the auxiliary quantum circuit and a value of the first measure of dispersion for the target estimator values; and whereinthe inference set comprises for each first error-mitigated target estimator value the associated value of the normalized measure of dispersion.

3. Method according to claim 2, whereininferring the final error mitigated target estimator value is on the basis of the first error-mitigated target estimator value with the smallest value of the normalized measure of dispersion among the plurality of first error-mitigated target estimator values.

4. Method according to claim 2, whereininferring the final error mitigated target estimator value comprises selecting a set of auxiliary quantum circuits among the plurality of auxiliary quantum circuits to thereby define a data set comprising for each auxiliary quantum circuit of the set a tuple of the derived first error-mitigated target estimator value and the associated value of the normalized measure of dispersion, defining a fitting model for the data set, fitting the fiting model to the data set and extrapolating to the limit of zero for the value of the normalized measure of dispersion to thereby infer the final error-mitigated target estimator value.P-00183WO 051220255. Method according to claim 4, whereinselecting the set of auxiliary quantum circuits comprises selecting only those auxiliary quantum circuits for which the associated value of the normalized measure of dispersion is below a predetermined threshold value.

6. Method according to anyone of the preceding claims, whereinthe noise-reduced target estimator value is derived in the noise-reduction step such that in the zero-noise limit the noise-reduced target estimator value is equal to the target estimator value, and whereininferring the first error-mitigated target estimator value in the inference step comprises estimating the zero-noise limit of the noise-reduced target estimator value with the optimal control parameter value.

7. Method according to anyone of the preceding claims, whereinfor each auxiliary quantum circuit deriving the first contribution of the noise- reduced target estimator value for each target noise-strength value comprises assigning one auxiliary noise strength value to the target noise strength value and dividing the target estimator value associated with the target noise-strength value by the auxiliary estimator value associated with the auxiliary quantum circuit and the associated auxiliary noise-strength value and by multiplying the result with the noise-free auxiliary value associated with the auxiliary quantum circuit.

8. Method according to anyone of the preceding claims, whereinfor each auxiliary quantum circuit the control parameter-dependent model for the remaining noise-dependency of the first contribution comprises a term which is a product of the control parameter and a noise-cancelling function in the noise strength which depends on the auxiliary estimator values and the noise-free auxiliary value associated with the auxiliary quantum circuit and has a noise-dependency which is reduced compared to the noise-dependency of the auxiliary estimator values associated with the auxiliary quantum circuit.

9. Method according to anyone of the preceding claims,wherein for at least one of the auxiliary quantum circuits the minimization criterion is fulfilled in the control parameter step when a zero-noise limit of a K-th Taylor polynomial of the noise-reduced target estimator value in the noise strength about an expansion value selected among the plurality of target noise strength values, wherein derivatives are replaced by difference quotients which are expressed in terms of theP-00183WO 05122025parameter-dependent noise-reduced target estimator values, is equal to the noise- reduced target estimator value associated with the expansion value, wherein inferring the first error mitigated target estimator value in the inference step is on the basis of the noise-reduced target estimator value associated with the expansion value and obtained for the optimal control parameter value, andwherein preferably the target noise strength values increase with the target noise level, and the expansion value is the smallest target noise strength value of the plurality of target noise strength values.

10. Method according to anyone of the preceding claims, whereinthe method comprises determining the noise-free auxiliary value for at least one auxiliary quantum circuit, wherein the determining comprises a classical simulation of the auxiliary quantum circuit at zero noise on a classical computer.

11. Method according to anyone of the preceding claims, whereinthe target and auxiliary quantum circuits are error-corrected quantum circuits according to a quantum error correction code, and the execution of the target and auxiliary quantum circuits at the plurality of different noise levels comprises their execution for a plurality of different code distances of the quantum error correction code.

12. Method according to anyone of the preceding claims, whereinthe execution of the target and auxiliary quantum circuits at the plurality of different noise levels comprises one of the following such as unitary folding on physical or logical qubits, local gate folding, noise amplification via pulse level control or probabilistic noise-amplification.

13. Method according to anyone of the preceding claims, whereinthe method further comprises executing the target quantum circuit and / or the plurality of auxiliary quantum circuits by the quantum processor at each of the plurality of target and auxiliary noise levels to thereby obtain the target measurement data and / or the plurality of auxiliary measurement data, respectively, and evaluating the target and / or auxiliary measurement data by the classical computer to thereby determine the target and / or auxiliary estimator values, respectively, and whereinpreferably the method comprises assigning one auxiliary noise strength value to each target noise strength value, and executing the target quantum circuit and the plurality of auxiliary quantum circuits by the quantum processor is such that for eachP-00183WO 05122025target noise strength value and the assigned auxiliary noise strength value, the target quantum circuit and all auxiliary quantum circuits of the plurality of auxiliary quantum circuits are executed consecutively at a target noise level associated with the target noise strength value and at an auxiliary noise level associated with the assigned auxiliary noise strength value, respectively..

14. Computer program product comprising instructions which, when the computer program product is executed by a classical computer, cause the classical computer to carry out the method according to anyone of claims 1-12.

15. Computing system comprising a classical computer and a quantum processor, wherein the quantum processor is configured to execute the target and auxiliary quantum circuits at each of the plurality of noise levels to thereby obtain the target and auxiliary measurement data, respectively,wherein the classical computer is configured to evaluate the target and auxiliary measurement data to thereby determine the target and auxiliary estimator values, respectively, and whereinthe classical computer is further configured to execute the computer program product according to claim 14.P-00183WO 05122025

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