Method of probabilistic noise shaping, quantum computer, classical computer, and system

Abstract

A method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit, wherein an upper limit of quantum circuit executions and an upper limit of quantum circuit shots is predetermined, the method comprising the steps of: considering a plurality of variants of the target quantum circuit that forms, together with the target quantum circuit, a set of quantum circuits, wherein each element of the set of quantum circuits is associated with a probability according to a probability distribution, identifying, from the set of quantum circuits and according to the probability distribution, a first set of quantum circuits and a second set of quantum circuits, sampling from the first set of quantum circuits according to a first sampling strategy, which is based on at least the upper limit of quantum circuit shots, a first list of quantum circuits, and executing the quantum circuits of the first list on the quantum processing unit, thereby obtaining a first list of results, sampling from the second set of quantum circuits according to a second sampling strategy, which is based on the upper limit of quantum circuit executions and the upper limit of quantum circuit shots, a second list of quantum circuits, and executing the quantum circuit of the second list on the quantum processing unit, thereby obtaining a second list of results, and performing post processing of the first and second list of results to obtain a result of the target quantum circuit subject to the probabilistic noise shaping.

Description

[0001] 277849 s27

[0002] Method of probabilistic noise shaping, quantum computer, classical computer, and system

[0003] Technical field

[0004] The present disclosure relates to probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit. It moreover relates to a quantum computer having a quantum processing unit configured to implement the probabilistic noise shaping, a classical computer having a classical processing unit configured to implement the probabilistic noise shaping and a system of the quantum computer and the classical computer.

[0005] Background

[0006] In the technical field of quantum computing, one of the more prominent, if not the most prominent model, is the quantum circuit (QC) model. This model can be seen as a quantum version of the classical circuit model used in classical computing in which classical bits are replaced by qubits (short for quantum bits) and quantum operations replace classical operations.

[0007] In a QC, the quantum operations may be represented by quantum gates. In general, a quantum gate (or simply gate) refers to a fundamental physical operation that manipulates the quantum state of one or more qubits. The skilled person understands that quantum gates may be represented by matrices and qubits (or more precisely: the quantum state of the qubit) may be represented by vectors or matrices, and the application to qubits involves matrix multiplications. When a quantum gate is applied to one or more qubits, it transforms the quantum state of those qubits. Performing a gate on one or more qubits may thus be considered as involving a manipulation of the corresponding quantum state (s) of the qubit (s) according to the specific operations defined by the gate.

[0008] However, quantum devices, in particular in the current era of Noisy Intermediate-Scale Quantum (NISQ) devices, are often subj ect to substantial noise leading to faulty quantum operations. While conceptually, quantum error correction by means of so-called error correction codes, is possible, at the present stage of development, the technical requirements for it to be feasible are not met yet. This is in particular due to the no-cloning theorem of quantum mechanics which requires quantum error correction to use more advanced techniques than usable in classical computation.

[0009] Quantum error mitigation, on the other hand, aims at mitigating, not outright removing the error. For this to be feasible, accurate knowledge about the source of the errors, that is, the noise, is necessary such that the mitigation techniques can function. While an accurate noise characterization can lead to meaningful quantum error mitigation, an imprecise or even incorrect noise characterization may lead to quantum error mitigation being less useful as a tool or even detrimental. Indeed, an imperfect error characterization of the noise has direct detrimental effects on the implementation of techniques like probabilistic error cancellation, in which noise is (completely) canceled, and / or probabilistic Zero Noise Extrapolation (ZNE), in which noise is partially canceled and / or amplified, where it can lead to over or under compensation of the noise.

[0010] Moreover, a precise characterization can help in improving calibration by providing detailed feedback, in identifying good and stable patches with best performing qubits and, in general, develop a better understanding of how the quantum processing unit (QPU) works. Accordingly, an accurate noise characterization is important for quantum computing, specifically for implementing the above-mentioned noise-aware error mitigation techniques as well as to boost error correction codes such that the quantum advantage of quantum computing outperforming (in certain tasks) classical computing can be achieved.

[0011] Many current quantum error mitigation techniques can be seen as probabilistic noise shaping, that is, techniques that effectively change and / or reshape the noise affecting a target quantum circuit (also referred to as "target circuit") to implement the error mitigation. One important aspect thereof is that these techniques require that several variants of the target quantum circuit have to be executed. A variant of a target quantum circuit may mean that some additional operations / gates are added to the target quantum circuit at specific position within the target quantum circuit.

[0012] Once all the required circuit variants are executed, the combined results can be post processed. The specific post processing procedure, as well as the nature and location of the added operations defining the variants may depend on the noise characterization of the target quantum circuit. If the latter is exact, one can show that the final result is equivalent to what one would have obtained from the execution of the sole target quantum circuit, subj ect to the desired reshaped noise.

[0013] Further, the variants of a target quantum circuit to be executed for a specific probabilistic noise shaping technique may also depend on the probabilistic noise shaping technique. ( In the following "probabilistic noise shaping" and "probabilistic noise shaping technique" may be used interchangeably. ) That is, depending on the goal of the probabilistic noise shaping technique, e. g., whether noise should be canceled or amplified, the variants may be different. Hence, it is the combination of probabilistic noise shaping technique and noise characterization that oftentimes defines the variants of the target quantum circuit that are, in combination with the target quantum circuit, the building blocks of the probabilistic noise shaping. For the sake of completeness, it is noted that (Pauli) twirling is an example of a method involving variants of a target quantum circuit that are created independently from the noise characterization.

[0014] Further, for many techniques of probabilistic noise shaping, the number of variants should ideally be very large. In fact, in theory, for many probabilistic noise shaping techniques achieving an ideal result requires an infinite number of circuit variants. Clearly, the number of (different) circuits that can be executed on a (current) quantum computer is limited, typically to the order of few thousands, in order to keep the total runtime of the experiment at reasonable levels.

[0015] In this context it is also important to note that the fragile nature of quantum systems, even when carefully engineered, may also limit the time available before a recalibration of the system is required. Accordingly, a subset of variants has to be determined that is executed when performing probabilistic noise shaping.

[0016] As a result, the implementation of noise reshaping techniques is limited, and their output will differ from the ideal one. For example, if the aim is to completely cancel the noise, because of the limited number of executable circuit variants, the final results will be different from what would have been obtained from a perfect noiseless execution of the target quantum circuit.

[0017] In this context, it is important to elaborate how circuits are performed on current quantum hardware and how this affects the total runtime. Before a circuit can be executed in a quantum hardware it has to be "loaded" into the system. Typically, batches of circuits are loaded. This loading time t; per batch takes relatively long, for example, may be in the order of a few seconds, while the loading time per individual circuit ti / B, with B being the batch size, may be in the order of tens of milliseconds. Once the circuit is loaded, it can be run several times, wherein each run (also referred to as "shot") produces an output, e. g., a bit string obtained from measuring the qubits. The time needed for each shot, ts, is typically much shorter than the loading time t;. In summary, the total runtime tris roughly the sum of the total loading time t - Nc / B, where Ncis the total number of (different) circuits loaded into the quantum hardware, and the total execution time ts- Ns, where Nsis the total number of shots, which are distributed in some manner among the Nccircuits.

[0018] Since typically ts« ti / B, one generally can afford NS» NC. A realistic scenario may be that Ncis of the order of thousands whereas Nscan then be in the order of millions. It is important to be aware of the orders of magnitude of these different budgets, that is, total number of (different) circuits and total number of runs (shots), as they define the "budget", i. e., the available resources, when implementing noise reshaping techniques.

[0019] Accordingly, one important aspect is how to select from the possibly infinite - but at the very least very large - number of circuits the circuits to be loaded into the quantum computer since there is an effective limit of Nccircuits, and, how many shots, of the total number of shots Ns, should be given to each circuit.

[0020] One approach, which is also the approach that has so far been widely used, is use Monte Carlo sampling, see for example arxiv. org / abs / 1612. 02058, as well as arxiv. org / abs / 2006. 12509, which may roughly be summarized as sampling the variants simply according to their weight in the probabilistic noise shaping technique and then assign a fixed number of shots Ns / Ncto each circuit variant. This approach oftentimes leads to satisfactory results and is generally well justified since the post processing of the results as part of the probabilistic noise shaping is typically linear and requires weighting each individual result coming from each circuit variant according to a probability (all of which depends on the noise characterization).

[0021] The established Monte Carlo approach provides a list of Nccircuits to load and execute obtained by randomly sampling the set of all possible variants {v (including the target circuit) according to a probability distribution {p. Note that the same circuit variant V can be sampled more than once, which is fine and even intended by Monte Carlo techniques. In this way, every variant ty will be executed on average rii = Nc- pi times (rounded up to the closest integer). Consequently, for large Ncthe result of the procedure converges to the ideal one, and in fact the mismatch between the ideal result and the experimental one converges to zero a

[0022]

[0023] s

[0024] As a remark, it is noted that many probabilistic noise shaping techniques require that every time a circuit variant v, is sampled, it undergoes Randomized Compiling to twirl the noise. In this common scenario, the circuit which is loaded is a recompiled version of v,, which is logically equivalent to the latter but can differ at the individual gate level. Within the context of Monte Carlo sampling, this means that if a given variant v, is sampled more than once (i. e., TI, > 1) and Randomized compiling is applied, this still corresponds to the loading of

[0025]

[0026] different circuits (albeit all logically equivalent to ty).

[0027] Some specific implementations of probabilistic noise shaping techniques, e. g., the ones based on sparse Pauli-Lindblad (SPL) noise characterization, allow, thanks to product structure of the effective noise model considered, for a convenient and efficient implementation of the above-mentioned Monte Carlo approach, which does not require the actual computation of the probabilities pt, see for example arxiv. org / abs / 22 01. 09866 and nature. com / articles / S41586-023-06096-3.

[0028] Further, as discussed below, one can treat Randomized Compiling (also referred to as Twirling) and the creation of the probabilistic circuits based on the noise characterization on the same ground: each one individually (as well as the two of them together) leads to a set S of circuit (potentially featuring subsets of logically equivalent circuits) whose result need to be post-processed together. In case of only Twirling, the post-processing will be a simple average (i. e., the weights are all the same). In case of only probabilistic reshaping, the weights of will be different real numbers, i. e., a weighted average. In case of Twirling and probabilistic reshaping, the post-processing will be a linear combination of these two cases.

[0029] It is important to note how shots are associated with each circuit execution. In the standard Monte Carlo approach, the same number of shots ns= Ns / Ncis associated with every circuit variant loaded on the hardware. This is indeed the best way to distribute the shots to minimize the impact on the final result of the statistical uncertainty due to the finite Ns, which is known as shot noise.

[0030] However, the distribution of the Nccircuits according to the conventional Monte Carlo sampling, also referred to as a tuple of Nccircuits, i. e., the list of elements being sampled in which one circuit may potentially appear more than once, can be far from optimal in some relevant practical scenarios, for example when the noise is weak and / or when the required amount of noise shaping is small ( for example in the case of small noise amplification as part of a ZNE strategy). Indeed, in these cases the probability distribution {p is very inhomogeneous, and a standard Monte Carlo sampling would then sample many times only a few circuits variants, leaving very little budget for the remaining circuit variants with lower probabilities.

[0031] For example, if one circuit VQ of the set of the target circuits and its variants would have a probability of PQ = 0.95, this would mean that 95% of the circuits loaded into the quantum computer and 95% of the shots would be used for this single circuit. If one now assumes that the total number of circuits is Nc= 1000, this mean that 950 slots are "wasted" loading circuits which are all logically equivalent to the same variant v0, while only the remaining 50 slots can be used to sample from the remaining variants. As typical probability distributions of probabilistic noise shaping (techniques) have far more than 50 elements, this means that many circuits that are relevant to the probabilistic noise shaping (technique) are not loaded.

[0032] Hence, while the Monte Carlo approach is advantageous in terms of minimizing the shot noise if the same amount of shots is granted for every circuit execution, it is clearly non-optimal if a large portion of the limited resource are used on a single circuit of the set of target quantum circuit and its variants.

[0033] It is emphasized that this scenario is very likely to occur in real life situations in which oftentimes either the noise of the target quantum circuit is weak and / or when the noise shaping is minimal, as in this case the probability associated with the target quantum circuit is typically close to one. In these cases, the standard MC strategy is thus extremely inefficient and better alternatives need to be developed.

[0034] In view of these existing approaches, there is a need for methods of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit that allow for a more efficient use of the ( finite) resources, thereby improving the precision of the noise shaping (when using the same amount of resources). Summary

[0035] The present disclosure has been made in view of the above technical limitations of currently existing methods of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit and thus provides a method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit that allows for a more efficient use of the ( finite) resources, thereby improving the precision of the noise shaping.

[0036] According to an aspect of the present disclosure, a method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit is provided, wherein an upper limit of quantum circuit executions and an upper limit of quantum circuit shots is predetermined, the method comprising the steps of: considering a plurality of variants of the target quantum circuit that forms, together with the target quantum circuit, a set of quantum circuits, wherein each element of the set of quantum circuits is associated with a probability according to a probability distribution, identifying, from the set of quantum circuits and according to the probability distribution, a first set of quantum circuits and a second set of quantum circuits, sampling from the first set of quantum circuits according to a first sampling strategy, which is based on at least the upper limit of quantum circuit shots, a first list of quantum circuits, and executing the quantum circuits of the first list on the quantum processing unit, thereby obtaining a first list of results, sampling from the second set of quantum circuits according to a second sampling strategy, which is based on the upper limit of quantum circuit executions and the upper limit of quantum circuit shots, a second list of quantum circuits, and executing the quantum circuit of the second list on the quantum processing unit, thereby obtaining a second list of results, and performing post processing of the first and second list of results to obtain a result of the target quantum circuit subj ect to the probabilistic noise shaping. According to a further aspect of the present invention, a quantum computer having a quantum processing unit is provided, wherein the quantum computer is configured to implement the above method of probabilistic noise shaping.

[0037] According to a further aspect of the present invention, a classical computer having a classical processing unit is provided, wherein the classical computer is configured to implement the above method of probabilistic noise shaping.

[0038] According to a further aspect of the present invention, a system of the above quantum computer and the above classical computer is provided.

[0039] In addition, preferred aspects of the present disclosure are defined in the dependent claims.

[0040] According to the invention, a more efficient probabilistic noise shaping can be realized, in particular without increasing the required resources.

[0041] Brief description of the drawings

[0042] Embodiments of the present disclosure, which are presented for better understanding the inventive concepts, but which are not to be seen as limiting the disclosure, will be described with reference to the figures in which:

[0043] Fig. 1 shows a flow chart of a method of probabilistic noise shaping;

[0044] Fig. 2 shows an illustration of a target quantum circuit and circuit variations;

[0045] Fig. 3 shows an illustration of a conventional sampling strategy; Fig. 4 shows an illustration of a sampling strategy according to the present disclosure; and

[0046] Fig. 5 shows a reduction factor in the number of circuits for sampling strategies according to the present disclosure.

[0047] Detailed description

[0048] As the present disclosure relates to the technical field of quantum computing, in particular to details of how to perform probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit, the following paragraphs will provide further details regarding the technology referred to and the terms used within this disclosure to facilitate the understanding of the present disclosure and the inventive concepts disclosed herein.

[0049] Quantum computing can generally be understood as technically implemented computing based on or exploiting quantum mechanical phenomena. Under certain conditions, in particular at small scales, classical theories of physical matter have to be replaced by quantum theories. One core element of these theories is that physical matter exhibits properties of both particles and waves. Quantum computing is built on the fact that leveraging this behavior can lead to what is a called a "quantum advantage": For some calculations, there exist quantum algorithms that outperform classical algorithms, i. e., algorithms performed on a classical computer, by a substantial margin, in some cases even exponentially faster.

[0050] However, quantum computing as understood at present may not replace classical computing in general and for every type of calculation but only for specific technical applications for which a quantum algorithm outperforming known classical algorithms is known. Typical examples thereof include Shor' s algorithm for finding prime factors of an integer showing an exponential speedup compared to known classical algorithms and Grover' s algorithm for an unstructured search showing a quadratic speedup compared to known classical algorithms, both having a wide range of possible applications. Further fields where quantum computing is expected to outperform classical computing is the field of quantum simulation, i. e., simulating a quantum system by using another quantum system governed by equivalent equations, originally proposed by Richard Feynman as well as specific optimization problems, in particular hybrid algorithms combining quantum computing aspects with classical optimization techniques. These two examples are followed by various industries as they could improve the performance and feasibility of many computationally very demanding tasks such as drug discovery and drug development, logistics as well as engineering.

[0051] At the same time, the fragile nature of quantum states leads to the possible computational advantage from quantum computing to be closely tied to a demanding engineering challenge as the quantum behavior of these states has to be preserved for a sufficient amount of time. Due to the presence of noise disrupting the quantum behavior, the number of operations that can be performed on a quantum computer are limited and as a consequence, large-scale algorithms cannot be realized on the currently available Noisy Intermediate-Scale Quantum (NISQ) devices, i. e., devices with non-negligible noise.

[0052] Further, detailed knowledge about the noise present during quantum computing may enhance the computational capabilities of quantum computing via techniques such a quantum error correction and / or quantum error mitigation. These techniques may allow to adjust the quantum computing process in view of the noise to counteract / mitigate the same and / or perform processing on the final result to correct for errors that occurred during the quantum computing process. In this manner, the capabilities of quantum computing can be enhanced and / or it can be ensured that said capabilities can be used to their fullest. As quantum computing originates in quantum physics, but relates to the field of computer technology, there is a need for a model or representation to bring quantum physics and computer technology together. The model currently most established is the QC model based on the classical circuit model. In the (classical) circuit model, a (classical) circuit is comprised of bits, having either the value 0 or the value 1, to which gates are applied. In the QC model, each of these elements is replaced by its "quantum version". An example of a part of such a quantum circuit is shown in Fig. 3, which will be described in more detail later.

[0053] The quantum version of the bit is the qubit (also referred to as quantum bit). It is, similar to a classical bit, a two-level (or two-state) system, however, a quantum-mechanical two-level system, possibly an effective two-level system. As a consequence of quantum physics, a qubit may be in any coherent superposition of both states 0 and 1 simultaneously.

[0054] A qubit or quantum bit may be considered as the basic unit of quantum information technology as well as the two-level quantum-mechanical system. It may refer to a physical qubit (that is, physically implemented qubit) and / or a logical qubit. As the present disclosure relates to noise characterization of a quantum processing unit (QPU), the focus will be on the physical qubit.

[0055] A quantum gate (or simply gate) may be considered as the basic quantum circuit operating on one or more qubits. Depending on whether one refers to the logical qubits or the physical qubits, the quantum gate may thus either refer to an operation on logical qubits, or to an operation in the context of the physical quantum-mechanical two-level system, i. e., a quantum gate operating on the physical qubits (as implemented by hardware, the physical quantum system).

[0056] Quantum gates may operate on a various number of qubits. If it operates only on one qubit, the gate is also called a "single- qubit gate". Accordingly, "two-qubit gates" operate on two qubits. While also gates operating on three or more qubits are possible, for most applications, only single- and two-qubit gates are used. Furthermore, any quantum algorithm can always be transpiled into single- and two-qubit gates.

[0057] A quantum circuit may be considered as a set of quantum gates, in particular a set of quantum gates spanning one or more layers.

[0058] A layer of quantum gates, as part of the quantum circuit, may herein be defined as (consisting of ) a set of quantum gates of the quantum circuit. It is possible that this set of quantum gates of the quantum circuit can be executed in parallel. Here, executing gates may mean that on the physical quantum systems operations are carried out that correspond to the execution of the quantum gates on the physical qubits.

[0059] A noise characterization of a quantum processing unit may be considered as the properties of quantum processing unit related to the noise the quantum processing unit is subj ected to. While the noise characterization may be considered to mainly relate to the quantum processing unit, it cannot be considered separately from the gates to be executed. Moreover, while such a noise characterization relies mostly on the individual (two qubit) gates, it may also depend on potential (spatial) cross talk and other phenomena. That is, the noise characterization may not necessarily depend only on the two qubits but may depend on the whole quantum circuit and the whole quantum processing unit.

[0060] At present, there are various possible platforms for quantum computing. One of the more prominent ones are super conducting circuits featuring one or more non-linear Josephson junctions used as the physical system of the (physical) qubits.

[0061] Using the above, the following describes embodiments of the present disclosure in detail. Fig. 1 shows a flow chart for a (computer-implemented) method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit, wherein an upper limit of quantum circuit executions and an upper limit of quantum circuit shots is predetermined, the method comprising the steps of: considering a plurality of variants of the target quantum circuit that forms, together with the target quantum circuit, a set of quantum circuits (S100), wherein each element of the set of quantum circuits is associated with a probability according to a probability distribution; identifying, from the set of quantum circuits and according to the probability distribution, a first set of quantum circuits and a second set of quantum circuits (S200); sampling from the first set of quantum circuits according to a first sampling strategy, which is based on at least the upper limit of quantum circuit shots, a first list of quantum circuits, and executing the quantum circuits of the first list on the quantum processing unit, thereby obtaining a first set of results (S300); sampling from the second set of quantum circuits according to a second sampling strategy, which is based on the upper limit of quantum circuit executions and the upper limit of quantum circuit shots, a second list of quantum circuits, and executing the quantum circuits of the second list on the quantum processing unit, thereby obtaining a second set of results (S400); and performing post processing of the first and second list of results to obtain a result of the target quantum circuit subj ect to the probabilistic noise shaping (S500).

[0062] Here, shaping may also include reshaping. In other words, " (noise) shaping" within the present disclosure may be considered to include any type of technique that modifies the noise present in a quantum circuit to be executed on a quantum processing unit.

[0063] Fig. 2 shows an illustration of a target quantum circuit and circuit variations, that is, may serve as an illustration of the set of quantum circuits. Specifically, Fig. 2 shows on the top left, as #0, the target quantum circuit comprising initial states, single- and two-qubit gates as well as measurement operations. This is the circuit to be performed on the quantum processing unit assuming a noiseless scenario, that is, every operation, i. e., every gate, is performed without any imperfections, that is, without any noise.

[0064] As this assumption is incorrect for the majority, if not all, circuits executed on quantum processing units, the probabilistic noise shaping is a technique that supplements the target quantum circuit (#0) with circuit variants, here shown as variants #l-#3. As can readily been seen, these circuits are indeed variants of the target quantum circuit in the sense that they share the same single- and two-qubit gates with the target quantum circuit, that is, generally are based on the target quantum circuit, but contain additional gates, here shown as solid black elements, in addition to the gates of the target quantum circuit. It is further noted that the variants shown here are not conclusive, rather there is also the possibility that in a variant a gate that is to be added to the target circuit is combined with a gate of target circuit to result in a gate not present in the target circuit. In this case, the variant does not share all the gates with the target circuit, but rather shares a substantial part (a major part) of the gates with the target circuit.

[0065] In other words, a variant of the target quantum circuit may be considered to be based the target quantum circuit and may contain (in addition to the gates of the target quantum circuit) additional gates or may contain gates replacing gates of the target circuit. These additional gates may be singlequbits and / or two-qubit gates.

[0066] The upper limit of quantum circuit executions may mean the number of different quantum circuits that can be executed (during one execution of the method). This upper limit may be limited by the time required to load a quantum circuit into the quantum processing unit, which may also be referred to as loading time. Central to the present disclosure may be, as also explained elsewhere within this disclosure, that in present quantum hardware, but also in most devices of the foreseeable future, this loading time is much longer than the runtime.

[0067] Specifically, in typical implementations at the current technology level, the loading time per circuit (which may be loaded in batches) may be in the order of tens or a few hundreds of milliseconds, while the runtime for each execution (shot) of a circuit is much shorter (of the order of one milliseconds, likely going to go down to the realm of only tens of microseconds). Hence, the upper limit of quantum circuit shots may be larger (much larger, two, three or even more orders of magnitude larger) than the upper limit of quantum circuit executions.

[0068] In particular, the following considerations describing the current technology level may underline this. In a current quantum processing unit, the minimum execution for a single execution / shot of a circuit is limited by a so-called reset time to be in the order of around 0.5ms. With a technology called "active reset", this number can be reduced to a few microseconds, depending on the circuit. Moreover, there also exist a more advanced technology called "unconditional reset", which allows the reset time to be reduced to a few tens of nanoseconds.

[0069] Further, as mentioned also elsewhere, circuits are typically loaded into the quantum processing unit in batches. The size of these batches depends on many factors such as electronics, the type of circuits and others, but currently this is of the order of around 100 circuits per batch.

[0070] Thus, a single shot currently takes of the order of one millisecond or less but may easily be in the realm of tens of microseconds, while the loading time per circuit may be around ten or a few hundreds of milliseconds. Consequently, both at least at present levels of technology, there are at least three orders of magnitude between the time required for a single shot and the loading time of a circuit.

[0071] The upper limit of quantum circuit shots may mean the total number of quantum circuits that can be run, that is, how many single runs of any quantum circuit can be carried out (during one execution of the method). This upper limit may be limited by the time required to run a quantum circuit on the quantum processing unit, which may also be referred to as run time.

[0072] A probability may mean that each value is positive, and the sum of all values / probabilities is equal to 1.

[0073] The first set and the second set may be disjoint, the union of the first and the second set may be the set of quantum circuits.

[0074] The probability associated to each element of the set of quantum circuits may remain associated to each corresponding element of the first / second set of quantum circuits.

[0075] The probability distribution may be a probability distribution associated with the probabilistic noise shaping and / or may depend on the target circuit and / or may depend on a noise characterization of the quantum processing unit.

[0076] The first sampling strategy may involve an upper limit for the number of quantum circuit executions.

[0077] Performing post processing may involve obtaining the result of the target quantum circuit as linear combination of the results associated to each element of the set S of quantum circuits weighted by its respective probability. This may take the form of a sum R

[0078]

[0079] =aiRi = C SiPiRi, where i identifies each element of the set S,

[0080]

[0081] are the weights, Riare the results, si= sgn(ai) is the corresponding sign, C = Σpi|ai| and, finally, pi= |ai| / C defines a probability distribution.

[0082] Note that each circuit in the set S may be unique and may differ from all the other elements in the set (as per definition of a set). However, in general, there might be subsets consisting of circuits that are logically equivalent and differ only at the level of individual gates. Such a structure of S is typically the result of the combination of probabilistic error shaping and twirling. The former, starting from a single target circuit c, produces variations of the target circuit by randomly inserting individual gates, leading to a set of circuits which are generally different from each other at the logical level. The latter, which is necessary, for example, to enforce the validity of the Pauli noise model, creates variations of a given circuit by adding / modif ying gates with the constraint of not changing the unitary associated with the whole circuit.

[0083] The above discussed concepts are now illustrated in connection with Figs. 3 and 4. While Fig. 3 shows an illustration of a conventional sampling strategy, Fig. 4 shows an illustration of a sampling strategy according to the present disclosure.

[0084] Specifically, both figures represent the same probabilistic noise shaping scenario, namely a case of probabilistic noise shaping of a target quantum circuit, the circuit 0 associated with a probability PQ = 0.9, gives rise to a first variant, the circuit 1 associated with a probability pi = 0.06 as well as remaining circuits that share the remaining probability of 0.04 (since the weights are assumed in the present case to correspond to a probability distribution). In other words, 96% of the weights are associated to the first two circuits 0 and 1 while the remaining 4% of the weights are associated to the remaining circuits. Further, in line with the above considerations, the resources available for the probabilistic noise shaping are as follows. There are 1 000 executions available, that is, 1 000 circuits can be loaded into and then run on the quantum processing unit. Further, a total number of 1 000 000 shots (runs) are available.

[0085] Fig. 3 now shows the conventional approach making use only of a single sampling strategy (a Monte Carlo approach) applied to the entire set of circuits. As a consequence, as shown, following the probabilities p,, on average 900 executions and 900 000 shots are used for the target quantum circuit 0 and on average 60 executions and 60 000 shots are used for the first variant 1. For the remaining circuits 2,3,..., only on average 40 executions and 40 000 shots are available. While this will ensure that, given the resources, the shot noise is minimized (since the shots are distributed according to the weights), this means that the vast majority (around 96%) of the circuits executions are used on just two circuit variants (variant 0 and variant 1) and only around 40 circuits execution are left to sample the rest of the circuits.

[0086] A consequence of this is that if these remaining circuits are more than 40 (which is often the case in practically relevant scenarios), only fractions of all circuits can be loaded and run, thus limiting the precision of the probabilistic noise shaping. It is further noted that since many probabilistic noise shaping techniques involve Randomized Compiling, it is preferable for the performance of these techniques if each variant is loaded more than once into the quantum processing unit as then logically equivalent versions of the variant that differ on the gate-level are loaded.

[0087] Fig. 4 shows how this situation is improved by using a method of probabilj stic noise shaping according to the present disclosure. As can be seen from the first two columns of the table as well as the distribution shown in the bottom, the plurality of variants of the target quantum circuit and the quantum circuit are the same and their weights are the same as well. This aspect corresponds to step S100 above.

[0088] Different from Fig. 3, the method of probabilistic noise shaping according to the present disclosure now identifies (step S200 above) a first set of quantum circuits, namely circuits 0 and 1, and a second set of quantum circuit, namely the remaining circuits. As can be clearly seen from Fig. 3, the two circuits 0 and 1 are identified according to the probability distribution as the two circuits with the largest probability are identified (selected).

[0089] Then, from the first set of quantum circuits sampling is performed according to a first sampling strategy (step S300 above), this sampling results in the first list of quantum circuits. The quantum circuits of this first list are then executed on the quantum processing unit, and a first list of results is obtained. Specifically, as can be seen from the table of Fig. 4, each of circuit 0 and circuit 1 are executed, that is, loaded, 50 times and then executed 900 000 and 60 000 times, respectively.

[0090] As a consequence, 900 executions remain for the remaining circuits 2, 3,.... That is, from the second set of quantum circuits sampling can now be performed according to a second sampling strategy (step S400 above), this sampling results in the second list of quantum circuits. The quantum circuits of this second list are then executed on the quantum processing unit, a second list of results is obtained.

[0091] Finally, post processing of the first list of results and the second list of results can be performed (step S500 above), and thus a result of the target quantum circuit subj ect to the probabilistic noise shaping can be obtained. In more detail, this step may entail, first, post processing of the numerous shots of each execution of each circuit, second, post processing of the numerous executions of each circuit, and, third, post processing of the circuits, that is, the target quantum circuit and the variants of the target quantum circuit, according to the weights and the details of the probabilistic noise shaping.

[0092] As a result, the use of this method allows for more different circuit variations to be sampled and then loaded into and then run on the quantum processing unit while using the same resources, thus leading to an improved precision of the overall probabilistic noise shaping.

[0093] Specifically, in the present scenario, the reduction of the standard deviation of the result of the probabilistic noise shaping of the target quantum circuit, and therefore the statistical error associated with the result can be estimated as

[0094] 50 + 50

[0095] 11000

[0096] = 4,74.

[0097] 1 — a0— cq

[0098] This may also be understood as meaning that 22,5 more resources would be necessary if the conventional approach is used to obtain the same result as achieved by the method according to the present disclosure.

[0099] In this context, it is important to note that the number of shots remains the same as compared to the scenario of Fig. 3, that is, the number of shots is determined according to the probabilities p, since in this way the shot noise can minimized (as done in the conventional approach). Further, it is noted that the circuits 2,3,... and their associated probabilities p2,p3>... may be subj ect to a rescaling as they are then to be considered as a rescaled probability distribution which is used to distribute the 900 executions and 40 000 shots among these circuits. In the present case, the probabilities p2,p3,... may thus be rescaled as (3, = a, / (l — a0— cq) and the executions and shots within this set of circuits may be distributed, using for example a Monte Carlo approach, according to this rescaled distribution.

[0100] It is noted that the number of executions per circuit according to the first sampling strategy may be chosen according to the probabilistic noise shaping and / or may be chosen to ensure that the performance of Randomized Compiling is not affected, which may be achieved if the number of executions is at least 10, preferably at least 20, more preferably at least 30. Generally, the number of executions needed to ensure that the performance of Random Compiling is not affected may depend on the circuit and the hardware, specifically may depend on the relevance of coherent errors on the quantum processing unit. In line therewith, a good calibration may reduce the number of executions necessary for this.

[0101] Finally, since the number of shots remains unaffected in this scenario, it is ensured that this approach is strictly better than the conventional approach since in both approaches the shot noise is minimized while the approach according to the present disclosure allows for a more efficient distribution of the executions, i. e., the circuits loaded into the quantum processing unit are selected in an improved manner.

[0102] In the following, further details of sampling strategies according to the present disclosure will be discussed. In order to better understand these strategies, the conventional approach will be discussed first. In the conventional approach, by sampling Nccircuits from the set S with the probability distribution {p, one obtains a tuple of indexes T =

[0103]

[0104] (t1;.. tWc), corresponding to the circuits to be executed on the hardware (possibly multiple times, if the corresponding index appears more than once).

[0105] If one calls r“ the stochastic variable associated with the result of a single shot of the circuit i and if one assumes that every circuit is assigned the same number of shots Ns(which is optimal in the context of Montecarlo sampling), the probabilistic estimator for the quantity of interest is

[0106]

[0107] where, for simplicity, we absorbed the sign s_i into the definition of

[0108]

[0109] . By computing the average over the circuits c and the shots s, one can see that the expectation value of r is indeed R, i. e., r)s)c= R. It can be easily shown that the variance of r, i. e., Var(

[0110]

[0111] r) = ((r2)} — ((r)}, reads

[0112] C2 / L

[0113] Var(r) =

[0114] NC\NS

[0115]

[0116] ( 1 )

[0117] where A= ^Var(r^)s^ is the variance of the measured result due

[0118]

[0119] to shot noise, averaged over all sampled circuits, and v = Var is the variance of the results of each circuit run, averaged over the different circuits, of the result that would be obtained with an infinite number of shots. This expression provides the uncertainty that characterizes the postprocessed experimental result r and only requires to know v and A or, alternatively, at least to have an estimate for them. Note that, if the result one is interested in is the expectation value of a Pauli observable, both v and A can be upper bounded by 1.

[0120] Based on these considerations, it can be shown that it can be convenient to partition the set S into two sets Soand S. It is noted that a partitioning into more subsets is also possible. Based on this partitioning, one can then associate a probability to each one of them, by simply summing over the probabilities p, of each set' s elements. This gives us PQ = Z ' ieSo Pi and Pi = 1 — PQ • One can then express R as a sum of two terms

[0121] UiRi =

[0122]

[0123] and obtain an estimator for it by sampling twice, once from the set of circuits Soaccording to the re-normalized probability distribution Pi / Po}, resulting in the tuple To and once from the set of circuits Si according to the probability distribution. Pi / Pi}' resulting in the tuple 7. The estimator then reads

[0124] CPk

[0125] r =

[0126]

[0127] k z=0,l iET (X

[0128] Using the result above, and the fact that circuit executions are independent, on can readily obtain the variance of r as

[0129] V

[0130] var(, I „2 V( liJ2 / Ak,

[0131] r)= C2J-W|-(k) +Vk

[0132]

[0133] k c ' s

[0134] (2 )

[0135] where the index k is used to indicate the two subsets k = 0, 1 (k

[0136] with different numbers of circuits Ncand different number of shots per circuits N^k\ Similarly, the index k has been added to the quantities Afcand vkas they can be different for the two subsets Soand Si. Assuming to have a fixed total budget for circuit executions Nc=

[0137]

[0138] and for the total number of shots NsOi=

[0139]

[0140] Z / c Wc(k), one can assess what is the best strategy to allocate resources between the two sampling procedures in order to minimize the variance of r. The result of this constraint minimization reads

[0141] wf’

[0142] (3)

[0143]

[0144] (4 )

[0145] corresponding to a minimal variance of

[0146] 2 _ 2 ■ (Zfc Pky / Vk) Var(r)min= C2

[0147] NcNgOt

[0148]

[0149] (5)

[0150] (k} (k}

[0151] It is noted that, in practice, both

[0152]

[0153] Ncand will need to be rounded to the closest positive integers. To limit detrimental effects stemming from this operation, it may be necessary for the values for

[0154]

[0155] and obtained in Eqs. (3 ) and (4 ) to be consistently greater than 1.

[0156] It is instructive to recover the standard result for the variance Eq. ( 1 ) from the expression associated with the separate sampling of the two partitions Soand S. This can be achieved partitioning the circuits and the (total) shots according to P^, i. e., with N = NcPk,

[0157] ~ Ntot

[0158] N™ = ~^Pk= N s*oii / Nc.

[0159]

[0160] Inserting these expressions into Eq. ( 2 ), one obtains

[0161] Varl h _ c2 Pk ^k 2J / C Pk ^k

[0162] V ar (J standard “L H^tot

[0163]

[0164] ( 6 )

[0165] which is equivalent to Eq. ( 1 ) with the identi fications

[0166]

[0167] = NCNS, v='lk kvk r and

[0168]

[0169] A= Pk k. The last two identi fications can be readily veri fied by noting that v / Nc= M2vk / NcPk') and =lk(Pk)2^k / (NioiPk). It is noted that the two results are identical when Afcand vkdo not depend on k, which is what would happen, for example, i f the partitioning of S into Si and Sowas done randomly.

[0170] By comparing Eqs. ( 5 ) and ( 6 ), one can estimate the amount of resources that the optimal strategy, i. e., strategies according to the present disclosure, would allow to save while still providing the same level of statistical uncertainty. In particular, one can derive the ratios

[0171]

[0172] and, indicating the theoretical reduction factor in the required resources, which read

[0173] , > 2J / C Pk^k

[0174] ’ (Xk Pk^kf

[0175] , _ Ek Pk&k

[0176] S'? —?

[0177]

[0178] tik pk^k)

[0179] I f Akand vkdo not depend on k, we simply obtainc=s= 1. However, i f one finds a partitioning of the set S that leads to a signi ficant fc-dependence, then the optimal strategy can lead to significant decrease of the resources needed to perform probabilistic error shaping.

[0180] Specifically, there are useful and realistic scenarios that naturally feature v0« v1and show the effect of the optimized sampling strategy. A prominent example in this respect is when set S features one single circuit associated with a large probability or, taking into account the need to implement also twirling, a subset of logically equivalent circuits So, associated with a large probability PQ. In this case, it is very likely for v0to be significantly smaller than v1. Indeed, the former is associated with executions of circuits that differ only at the individual gate level, due to potential twirling, but which are overall logically identical and thus likely lead to very similar results. In this respect, a major contribution to the standard deviation

[0181]

[0182] may be the presence of coherent errors, which may not be expected to be dominant in a well-calibrated device. By contrast, v1is associated with the execution of circuits that differ from each other also at the logical level and thus give rise to very different results. As an example, two circuits that differ because of a bit-flip located just before the measurements can be associated with results with opposite sign. If one defines 7] =

[0183]

[0184] yjvx / vo as the ratio between the two standard deviations, one obtains the expression

[0185] (.=_ Pp(l - 7]2) + 7]2_

[0186]

[0187] cP0(l +772 —2?]) — 2P0(T]2—?]) + 7]2

[0188] which is plotted in Fig. 5. That is, Fig. 5 shows a reduction factor in the number of circuits as a function of PQ and r| for sampling strategies according to the present disclosure. In detail, Fig. 5 shows on the horizontal axis r|, on the vertical axis PQ, and the curves indicate the reduction factor in the number of circuits Nc, i. e.,. Herein, a reduction factor of 2 means that half the resources are needed to obtain the same uncertainty. Quite generally, one can observe that the larger r| and the larger Po, the larger the reduction factor.

[0189] In realistic scenarios, also currently observed in experiments, g ~ 10, the sampling strategies according to the present disclosure easily allow to halve the necessary resources for large values of PQ^O. 75. This gain is even more substantial if the twirling-related variance is smaller, i. e., when g > 10.

[0190] Finally, it is noted that

[0191]

[0192] should not be too small, in order to avoid substantial deviation from the expected performance once is rounded to the nearest integer. Assuming for simplicity A0~ Ax, this requirement is satisfied when NsOi / Nc» T] / (P0+ pPi) ■ Given the fact that in current quantum processing hardware, in particular superconducting quantum processing hardware, the time needed to perform a single shot is several orders of magnitude shorter than the one needed to load a circuit, this condition can be easily met.

[0193] Herein, the various method steps may also be included, that is, may be understood as, "sending instructions to a quantum processing unit to carry out the necessary steps". That is, the method of probabilistic noise shaping may relate to a method in which the steps are performed on a quantum computer comprising the quantum processing unit, or a method in which the steps are performed partially on a quantum computer comprising the quantum processing and partially on a classical computer, or a method in which all steps are performed on a classical computer, these steps being limited to instructing a quantum computer comprising the quantum processing unit to perform experiments to provide result and evaluating these results to obtain the noise characterization.

[0194] In other words, since the probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit can be seen as a result of data processing measurements results obtained by performing experiments on the quantum processing unit, the present invention does not necessarily involve the quantum processing unit but may also be implemented through interacting with the same by sending instructions / input to the quantum processing unit and receiving results / output of, for example, the first / second process, from the quantum processing unit.

[0195] In the following, further preferred embodiments are described. In a preferred embodiment, the probabilistic noise shaping may be one of: complete noise cancellation, partial noise cancellation, noise amplification, partial noise amplification or noise alteration.

[0196] An example of noise shaping may be given by Jose D. Guimaraes et al.: " Noise-Assisted Digital Quantum Simulation of Open Systems Using Partial Probabilistic Error Cancellation", PRX Quantum 4, 040329 (2023). An example of complete noise cancellation may be Probabilistic Error Cancellation (PEC). An example of partial noise cancellation may be a concept such as partial PEC, for example, PEC supported by symmetry-based post selection. An example of noise amplification may be probabilistic Zero Noise Extrapolation (ZNE). An example of noise alteration may be such as biasing the noise towards dephasing.

[0197] In a preferred embodiment, the first set of quantum circuits may comprise the target quantum circuit, In other words, the target quantum circuit may be subj ect to the first sampling strategy.

[0198] In a preferred embodiment, the plurality of variants may be based on the probabilistic noise shaping of the target quantum circuit. In other words, the variants may depend on the probabilistic noise shaping and / or the target quantum circuit.

[0199] This can be readily understood when considering that probabilistic noise shaping includes concepts such as PEC which aim at removing the noise as well as concepts such as Noise Amplification which aims at increasing the noise (oftentimes as part of ZNE such that the amplified noise is used to extrapolate the zero-noise result). Accordingly, it is clear that depending on the goal of the probabilistic noise shaping, the variants of the target quantum circuit differ.

[0200] In a preferred embodiment, the plurality of variants is based on a noise characterization of the quantum processing unit.

[0201] This can be readily understood when considering that depending on the noise present in the system, but also depending on the quality of the information available about the noise present in the system, shaping the noise takes a different form. Clearly, when the noise should be cancelled, such as if the probabilistic noise shaping is PEC, both the noise present in the system and the quality of the information available about the noise present in the system has an influence on how this is realized. Similar considerations apply to other types of probabilistic noise shaping, such as ZNE.

[0202] In a preferred embodiment, executing a variant of the target quantum circuit may involve recompiling the variant every n-th time it is run, with n being a natural number. Here, n may be 1, in which case the variant would be recompiled every time it is run.

[0203] As also noted elsewhere in this disclosure, the recompiled variant is logically equivalent to the variant. This may, for example, correspond to Randomized Compiling being a part of the probabilistic noise shaping.

[0204] Recompiling and executing the variant more than once may depend on the probabilistic noise shaping, in particular whether the probabilistic noise shaping involves Randomized Compiling.

[0205] In particular for probabilistic noise shaping, which includes Randomized Compiling, it can be readily understood that attributing more than one execution per element of the set of circuits is advantageous, specifically for the performance of the Randomized Compiling within the probabilistic noise shaping.

[0206] In a preferred embodiment, the first set may comprise not more than a predetermined number of elements of the set of quantum circuits or not more than a predetermined percentage of elements of the set of quantum circuits.

[0207] In a further preferred embodiment, the elements of the first set may be selected according to their associated probability.

[0208] In a still further preferred embodiment, the elements of the first sets may be selected according to their associated probability in a decreasing order.

[0209] In a preferred embodiment, the first sampling strategy and the second sampling strategy may be different.

[0210] In a preferred embodiment, the first sampling strategy may be deterministic and / or the second sampling strategy may be probabilistic. In other words, the first sampling strategy may have a deterministic character and / or the second sampling strategy maybe have a probabilistic character.

[0211] That is to say, the underlying concept of the first sampling strategy may be a deterministic one, while the underlying concept of the second sampling strategy may be a probabilistic one.

[0212] Here, it is noted that the presence of some elements that are randomized in a sampling strategy do not make the sampling strategy per se probabilistic are the way the sampling is done may nevertheless be deterministic. In other words, the presence of, for example, Randomized Compiling in a probabilistic noise shaping, does not make the sampling strategy used for such a probabilistic noise shaping probabilistic. Further, the character of the sampling strategy as "deterministic" rather than "probabilistic" may also be partial. Specifically, the sampling strategy may be deterministic with respect to the executions, but probabilistic with respect to the shots. This is to be understood as "deterministic" or at least as "partially deterministic".

[0213] In a preferred embodiment, the first sampling strategy may include limiting the number of executions per element of the first set to a predetermined threshold. That is, the number of executions may be independent of the probability associated to the element.

[0214] In a preferred embodiment, the number of shots per element of the first set may be determined according to the associated probability. Thus, the shots may be distributed according to the probabilities, thus achieving the desirable statistics also obtained when applying Monte Carlo sampling to the executions and shots.

[0215] However, since the executions are (typically) limited, this leads to more of the predetermined number of quantum circuit executions being available for the second set, thus more variants of the quantum circuit can be executed. This may in particular be advantageous with respect to shot noise.

[0216] In a preferred embodiment, the second sampling strategy may be adjusted according to the first sampling strategy.

[0217] In a preferred embodiment, the second sampling strategy may be on Monte Carlo sampling. This may be understood as using a conventional approach for the second sampling strategy.

[0218] Specifically, as also explained else in this disclosure, in particular in connection with Fig. 3, this may mean that the elements of the second set are sampled according to their probability, leading to a corresponding number of executions of this element and accordingly also to a corresponding number of shots of this element, which is the number of executions of this element times the ratio between number of quantum circuit shots and number of quantum circuit executions. This approach is unbiased and automatically distributes the shots to the probabilities.

[0219] In a preferred embodiment, the probabilities of the elements of the second set are rescaled in accordance with the first sampling strategy. This is, for example, also discussed above in connection with Fig. 4.

[0220] In a preferred embodiment of the present invention, parameters of the first sampling strategy are adapted during sampling from the first set and executing quantum circuits from the first list, and / or parameters of the second sampling strategy are adapted during sampling from the second set and executing quantum circuits from the second list.

[0221] Specifically, with reference to the above analysis, in order to come up with the optimal resource distribution, it may be advantageous to know the quantities

[0222]

[0223] and vk, i. e., parameters of the sampling strategies, beforehand. While this task can be challenging, good estimations for these quantities can be obtained during the execution of the experiment, thus allowing to adapt the strategies during the execution. For example, one can start guessing reasonable values of these parameters and then run the experiment with a small portion of the whole resources. By analyzing the data, it is straightforward to significantly refine the initial estimation and thus adapt the strategy so that, overall, the entire budget of resources is distributed in the optimal way.

[0224] In a further preferred embodiment of the presented invention, there is provided a quantum computer having a quantum processing unit, the quantum computer configured to implement a method according to an embodiment of the present invention. This quantum computer may, for example, be able to receive, as an input, a target quantum circuit to be execute by the quantum computer, a noise characterization and / or the probabilistic noise shaping to be applied and then perform, based on appropriately defined routines in line with the above, a method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit. In other words, the quantum computer can perform, given the quantum circuit, the method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit, that is, one may say that the quantum computer performs method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit autonomously.

[0225] There are no particular limitations regarding the possible hardware platforms on which the present invention may be implemented. Quantum processing unit based on superconducting circuits / superconducting qubits are a clear use case, in particular as the above cited papers have demonstrated that conventional cycle benchmarking can be realized on this platform. Nevertheless, an adaptation to any platform such spin qubits, ultra-cold atoms and the like is possible as the above discussion is platform independent.

[0226] In a further embodiment according to the present invention, there is provided a classical computer having a classical processing unit, the classical computer configured to implement a method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit according to an embodiment of the present invention.

[0227] This corresponds to the above discussion that the present invention may be implemented by a classical computer controlling a quantum computer, providing input to the quantum computer and receiving out from it. In this scenario, the quantum computer could be seen as an external (technical) entity controlled by a classical computer. In a still further embodiment according to the present invention, there is provided a system of a quantum computer according to an embodiment of the present invention and a classical computer according to an embodiment of the present invention. In other words, this system comprises the quantum computer as well as the classical computer. In such a system, the various steps involved in probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit may be shared and / or distributed among the classical computer and the quantum computer. They may in particular be distributed in a manner that allows efficient performance of said steps.

[0228] To underline the embodiments of the present disclosure, simulations of these embodiments have been performed. Specifically, several numerical simulations considering the execution of Clifford circuits on a 20-qubit quantum processing unit with a realistic noise profile. In particular, the following three different scenarios were considered to demonstrate that the present invention is applicable across the whole range of probabilistic noise shaping techniques:

[0229] a) Probabilistic noise amplification by a factor of 1.2 of a circuit with original success probability (i. e., the probability of executing it without errors) of 61%. b) Partial probabilistic noise cancellation (70% of the original noise) of a circuit with original success probability of 51%.

[0230] c) Complete probabilistic noise cancellation of the same circuit as in b).

[0231] In all cases, the same high-weight Pauli observable was measured and a budget of 4·106total shots was used. Further, two different values of the standard deviation associated with Pauli twirling, i. e., was considered. The standard sampling approach using a total number of circuits Nc= 1.4·104was performed and the result of the procedure was simulated, where the statistical uncertainty was estimating using bootstrapping (over both the shots and the circuits). The below reported results show that the same level of statistical uncertainty can be achieved with less circuits if strategies according to the present invention are implemented. As can be seen from the table below, depending on the scenario and the value of √v0, strategies according to the present invention require between 1. 6 and 4.7 times less circuits to achieve a comparable result.

[0232] Problem P0Nstot√v0Strategy NcResult

[0233] ||

[0234] Invention 4000 0.5268 ± 0.0017 fl Invention 3000 0.5278 ± 0.0016 ii Invention 6000 0.5835 ± 0.0026

[0235] Invention 4500 0.5881 ± 0.0026

[0236] Invention 8500 1.008 ± 0.011

[0237]

[0238] Invention 7500 0.989 ± 0.011

[0239] In summary, the present invention provides methods of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit by applying at least two sampling strategies involved in the probabilistic noise shaping. In this manner, the available resources can be used in a more efficient manner, that is, in a manner that avoid using a majority of the executions (majority of the circuits loaded) for a small number of circuits that are part of the probabilistic noise shaping.

[0240] Consequently, the probabilistic noise shaping provided by the present invention allows a better distribution of the available resources over the various circuits to be executed and run. In specific, relevant examples this can reduce the errors associated with results obtained by probabilistic noise shaping by a factor of 2 or more, which corresponds to using one quarter or less of the resources needed by conventional approaches to achieve the same level of precision.

[0241] Advantageously, methods of probabilistic noise shaping according to the present disclosure achieve this while maintaining the minimized shot noise achieved by the conventional approach such that the overall performance of the presented methods are a strict improvement over the conventional approaches.

[0242] Further, the present invention is not particular limited to a quantum processing unit topology, a quantum processing unit plat form / technology or the like, and thus can find application across the whole spectrum of quantum computing technologies, in particular can implemented whenever probabilistic noise shaping is made use of.

Claims

Claims1. A method of probabilistic noise shaping of a target quantum circuit to be performed on a quantum processing unit, wherein an upper limit of quantum circuit executions and an upper limit of quantum circuit shots is predetermined, the method comprising the steps of:- considering a plurality of variants of the target quantum circuit that forms, together with the target quantum circuit, a set of quantum circuits,wherein each element of the set of quantum circuits is associated with a probability according to a probability distribution,- identifying, from the set of quantum circuits and according to the probability distribution, a first set of quantum circuits and a second set of quantum circuits,- sampling from the first set of quantum circuits according to a first sampling strategy, which is based on at least the upper limit of quantum circuit shots, a first list of quantum circuits, and executing the quantum circuits of the first list on the quantum processing unit, thereby obtaining a first list of results,- sampling from the second set of quantum circuits according to a second sampling strategy, which is based on the upper limit of quantum circuit executions and the upper limit of quantum circuit shots, a second list of quantum circuits, and executing the quantum circuit of the second list on the quantum processing unit, thereby obtaining a second list of results, and- performing post processing of the first and second list of results to obtain a result of the target quantum circuit subj ect to the probabilistic noise shaping.

2. The method according to claim 1, wherein the probabilistic noise shaping is one of: complete noise cancellation, partial noise cancellation noise amplification, partial noise amplification or noise alteration.

3. The method according to claim 1 or 2, wherein the first set of quantum circuits comprises the target quantum circuit.

4. The method according to any one of claims 1 to 3, whereinthe plurality of variants is based on the probabilistic noise shaping of the target quantum circuit,and / orthe plurality of variants is based on a noise characterization of the quantum processing unit.

5. The method according to any one of claims 1 to 4, wherein executing a variant of the target quantum circuit involves recompiling the variant every n-th time it is run, with n being a natural number.

6. The method according to any one of claims 1 to 5, wherein the first set comprises not more than a predetermined number of elements of the set of quantum circuits and / or not more than a predetermined percentage of elements of the set of quantum circuits.

7. The method according to claim 6, wherein the elements of the first set are selected according to their associated probability, preferably in a decreasing order.

8. The method according to any one of claims 1 to 7, whereinthe first sampling strategy and the second sampling strategy are different,and / orthe first sampling strategy is deterministic and / or the second sampling strategy is probabilistic,and / orthe first sampling strategy includes limiting the number of executions per element of the first set to a predetermined threshold.

9. The method according to any one of claims 1 to 8, wherein the number of shots per element of the first set are determined according to the associated probability.

10. The method according to any one of claims 1 to 9, wherein the second sampling strategy is adjusted according to the first sampling strategy.

11. The method according to any one of claims 1 to 10, wherein the second sampling strategy is based on Monte Carlo sampling,wherein preferably the probabilities of the elements of the second set are rescaled in accordance with the first sampling strategy.

12. The method according to any one of claims 1 to 11, whereinparameters of the first sampling strategy are adapted during sampling from the first set and executing quantum circuits from the first list,and / orparameters of the second sampling strategy are adapted during sampling from the second set and executing quantum circuits from the second list.

13. A quantum computer having a quantum processing unit, the quantum computer configured to implement the method according to any one of claims 1 to 12.

14. A classical computer having a classical processing unit, the classical computer configured to implement the method according to any one of claims 1 to 12.

15. A system of a quantum computer according to claims 13 and a classical computer according to claim 14.

Images