Method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device

Abstract

A method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device, said quantum processing device including a plurality of n ≥ 2 qubits, means for applying a set G of quantum gates and means for applying measurements to the n qubits is disclosed, said method comprising: for each of a plurality of different circuit depths m ≥ 1, generating a plurality of quantum circuits C m , each comprising m sets of randomly sampled instructions with gates of said set (A) of quantum gates, for each quantum circuit C m of circuit depth m applying said quantum circuit C m to a random initial state p v of the n qubits to thereby obtain an output state pr thereof, and ρc m,V applying said measurement means to the n qubits so as to provide an outcome of a randomized measurement of said output state; calculating, by classical postprocessing of the measurement outcomes, an estimator value (B) of an average purity of all output states ρc m,V of the circuits of depth m on the basis of the measurement outcomes of the randomized measurements; conceiving a fitting model for a data set {(m, (B))} of the estimator values of the purity (B) and the circuit depth m, wherein a unitarity ū of the noisy gates of the randomly generated circuits C m is a parameter of said model, said unitary being indicative of the average contribution of the coherent noise; fitting the model to the data set to thereby obtain an estimation of the value of the unitarity ū.

Description

[0001] Method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device

[0002] The present invention is related to a method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device.

[0003] One of the main obstacles for quantum computing is the presence of noise, causing a multitude of errors that limit the usefulness of currently-available quantum processors. As not all quantum noise is made equal, figures of merit such as the fidelity of a quantum operation are by definition insensitive to many of the details of such noise.

[0004] Quantum noise can be described as an undesirable quantum transformation mapping the ideal noiseless output state of a quantum computation into some other valid, albeit unexpected, output quantum state. Once assuming that such noise is Markovian, i.e., it only depends on the input state and no other external context variables, a comprehensive classification of all possible errors can be made [1], but in particular it is extremely important to be able to distinguish between coherent and incoherent error contributions.

[0005] Coherent noise can be described as a deterministic, undesired unitary transformation, e.g., it can arise due to imperfect control or calibration of quantum gates, and also due to certain types of crosstalk [2]. Incoherent or stochastic noise, on the other hand, is described by non-unitary transformations and arises purely statistically, e.g., a bit-flip occurring with a certain non-zero probability.

[0006] The reasons why distinguishing between coherent and incoherent error contributions is important, include: i) Coherent errors can be suppressed by good experimental quantum control, while incoherent errors in isolation can be efficiently mitigated [3-7] and fully corrected [8, 9], ii) Coherent errors can accumulate in a much more detrimental way and provide error-rate estimates, such as average gate-fidelity, that can differ by orders of magnitude from its worst-case counterparts (i.e., fault-tolerance thresholds) [10-14], iii) The interpretation of average gate fidelity as a reliable and meaningful error rate depends heavily on the noise having a low coherent contribution [12, 15-17], The average coherence of a noisy quantum gate can be understood as a measurement of the rate at which noise shrinks the n-dimensional Bloch ball, (as depicted in Fig. 1 and described in more detail below), and can be measured by its unitarity, which in turn can be estimated operationally by Unitarity Randomized Benchmarking (URB)

[0012] . Despite being efficient and robust to State Preparation and Measurement (SPAM) errors, URB is not scalable in system size, as it requires estimating an exponential amount of expectation values within a Randomized Benchmarking (RB)-like protocol. (So-called Speckle Purity Benchmarking is a technique that can indirectly estimate average unitarity for larger systems; however it is not a standalone technique but a byproduct of Cross-Entropy Benchmarking

[0018] , which itself is famous for precisely requiring classical computation scaling exponentially in system size, and serving rather as a fidelity proxy under certain conditions [19, 20]). The reason for this is not URB per se, but rather that fundamentally, the unitarity is a second-order functional of the noise, akin to the case of the so-called purity with respect to a quantum state.

[0007] In general, the complexity of characterizing quantum noise in a comprehensive way typically scales exponentially with the system size [21, 22], Even most efficient protocols estimating a single figure of merit, such as those within the family of RB

[0023] , fail to do so at scale. However, with the quantum industry inevitably moving towards larger systems, there is a pressing need to find ways to generalize such tools to larger systems.

[0008] Recently, scalable RB-based techniques such as Mirror Randomized Benchmarking (MRB) [24, 25] and Binary Randomized Benchmarking (BiRB)

[0026] , among others [27, 28], have been developed, which are able to estimate the average fidelity of sets of quantum operations in a scalable, efficient, and SPAM-robust way, for a large class of gate sets. On the other hand, Randomized Measurements (RM) techniques, and so-called classical shadows, have enabled the efficient estimation of properties of many-body quantum systems [29, 30] (including state fidelity and purity), linear properties of gate sets

[0031] , and characterization of processes with memory

[0032] .

[0009] Due to the problems in the prior art, it is therefore an object of the present invention to provide a method for efficiently estimating the average contribution of coherent noise in an operation of a quantum processing device and to provide a computing system to carry out said method.

[0010] According to a first aspect of the present invention, there is provided a method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device, said quantum processing device including a plurality of n ≥ 2 qubits, means for applying a set G of quantum gates and means for applying measurements to the n qubits, said method comprising: for each of a plurality of different circuit depths m>1 , generating a plurality of quantum circuits Cm, each comprising m sets of randomly sampled instructions with gates of said set G of quantum gates; for each quantum circuit Cmof circuit depth m applying said quantum circuit C to a random initial state pvof the n qubits to thereby obtain an output state prthereof, and applying said measurement means to the n qubits so as to provide an outcome of a randomized measurement of said output state; calculating, by classical postprocessing of the measurement outcomes, an estimator value of an average purity of all output states pCm vof the circuits of depth m on the basis of the measurement outcomes of the randomized measurements; conceiving a fiting model for a data set {(m, of the estimator values of the purity and the circuit depth m, wherein a unitarity u of the noisy gates of the randomly generated circuits Cmis a parameter of said model, said unitary being indicative of the average contribution of the coherent noise; fitting the model to the data set to thereby obtain an estimation of the value of the unitarity u.

[0011] The present application was originally filed in color, and any reference to color in the present copy relates to the colors as originally filed.

[0012] The method according to the present invention provides an efficient method for estimating the average contribution of coherent noise of a quantum processing device. The method is based on classical postprocessing of measurement outcomes of randomized measurements on output states obtained by applying randomly generated quantum circuits to random initial states, which may be efficiently implemented on a classical computer.

[0013] It is well known that no physical system can be perfectly isolated. In particular, the loss of coherence of quantum states is one of the fundamental practical problems facing any quantum technology. In general, the dissipation of information to an external environment results in increased subjective uncertainty as to what the quantum state of a system is, which can be quantified by the purity of the respective density matrix p, given by tr[p2]. The purity takes extremal values of 1 for a pure state, and 2’nfor a maximally mixed state, and it can also be written as the Schaten 2-norm ||p II2. where Hxlll = tr[XX+], Logical operations or gates Q in a quantum computer / quantum processing device are theoretically described by unitary operators, which, by definition, preserve the purity of quantum states, i.e. , for pout= G(pin), we have tr[poMt] = tr[p?n]. In particular, a k-qubit gate is theoretically described by a unitary transformation on the k-qubits to which the k-qubit gate is applied. The set G of quantum gates comprises a plurality of k-qubit gates, wherein different gates may act on different numbers of qubits.

[0014] The quantum processing device includes means for applying the set G of quantum gates. Instructions with this gate set on the n qubits are also referred to as “layers” within this application, and also in the literature on RBM, see e.g.

[0025] , In the following, L(G) denotes the corresponding set of layers. In practice, a rule on how to apply the sampled gates must also be specified, depending, e.g., on the topology, connectivity, and the desired density of gates.

[0015] The quantum circuit Cmcomprises m sets of randomly sample instructions Ltwith gates of said set G of quantum gates. Each set of instructions Ltmay comprise one or more subsets of instructions with gates from the gate sets, the subsets of instructions being subsequently applied. Specifically, each of the sets of randomly sampled instructions may be in the form of a composite layer composed of a pair of subsequent individual layers composed of gates performed in parallel. In particular, the composite layer L may be of the form L, = wherein the individual layer lf^ is performed after the individual layer

[0016] The sets of instructions are randomly sampled, in particular according to a user-specified distribution Q. The quantum circuits Cmmay be generated by use of a classical computer which randomly samples the layers Ltaccording to the distribution Q.

[0017] According to the method of the present invention, a plurality of quantum circuits C is generated for a plurality of different circuit depths m. That is, there are at least two, preferably more different circuit depths. In one example, m = 1 , 2, 5, 10, 20, 80, but the invention is not limited to this. The circuit depth m is the number of layers Lt in the circuit C.

[0018] For each of these circuit depths m, a plurality of NCmquantum circuits Cmis generated. The number NCmof quantum circuits may be the same for each circuit depth m in one example. In another example, the number NCmof quantum circuits may be different for different circuit depths m.

[0019] According to the present invention, n-qubit random quantum circuits (7) of layer circuit depth m are generated. The operation is read (or acts on the qubits) from right to left. The layers Ltare randomly distributed according to the probability distribution Q. The term depth or circuit depth implicitly means layer circuit depth, or number of layers entering the given circuit. That is, depth is meant in a logical sense, that is before transpilation to a given basis gate set.

[0020] For each quantum circuit Cm of circuit depth m there may be one random initial state pv, said random initial state being different for different quantum circuits Cm. Alternatively, there may be for each quantum circuit Cm a plurality of different random initial states. That is, each quantum circuit Cmis applied to each of the random initial states of the plurality. Preparation of the random initial state may also be subject to noise. Applying the quantum circuit Cmto the initial states means applying the gates from the gate set G according to the instructions / layers of the quantum circuit by the quantum processing device. To this end, the quantum processing device may comprise an input for receiving the instructions of the quantum circuit, and the method may also comprise providing the quantum circuits to the quantum processing device.

[0021] Given an ideal operation g (e.g., one of the quantum circuits Cm), which may in theory (and in particular in a noiseless scenario), be described by a unitary operation, the corresponding, real noisy operation, gnoisy, is described by a more general Completely Positive (CP) map, such that for pout= gnOisy(Pin)> we have tr[poUtl tr [pfj. The reduction in purity of states when acted on by a noisy channel is related to the unitarity of such channel, a measure of “how far” gnoisyis from being unitary. Because locally we can always relate a noisy channel with an ideal gate by gnoisy := £°g (see Ref. [1]) for some CP map £, the unitarity of the noisy gate equivalently measures how far £ is from being unitary.

[0022] The average unitarity of a quantum channel may be defined as in

[0012] , by where £'(■) ■= £(• -1 / 2”) with 1 being an n-qubit identity operator, and the normalization 2" / (2"-1) ensuring that 0 ≤ ii(£) ≤ 1 , and with u(£)= 1 if and only if £ —and hence the noisy operation gnoisy— is unitary. The definition in Eq. (1 ) implies that the average purity of a noisy state is a combination of average unitarity and trace-decrease and / or non-unital terms. The unitarity only quantifies how unitary is, but not whether it is the target unitary, i.e., it does not measure whether ^noisy is far from the ideal £. A figure of merit aiming to quantify this distinction is the average gate fidelity, where implicitly we denote F(£) := F(£,!I), the average gate fidelity of £ with respect to the identity X (and in slight abuse of notation ip = IVOW'D- The average gate fidelity satisfies l / (2n+ 1) ≤ F(£) ≤ 1 with F(£) = 1 if and only if £ = T (equivalently if and only if ^noisy= G)- With respect F, the average unitarity satisfies: as shown in

[0012] , with saturation occurring for a depolarizing (purely incoherent) channel.

[0023] The unitarity can also be writen in terms of the fidelity as u(8) = (2n / (2n- l))F(£'+£'), where £'+is the adjoint map of 8'. Thus, the average unitarity can also be understood as a second-order functional of the fidelity of £', quantifying on average how distinguishable is the composition £'+£' from the identity. Under certain assumptions (namely, that £ approximately models the average noise of a gate set G in a temporally-uncorrelated (Markovian), time- independent and gate-independent way), F(£) can be estimated through RB and the unitarity u(£) can be estimated through URB [12,26]. Both techniques are not scalable in number of qubits, however, recently the MRB and BiRB variants were developed in [24 - 26], allowing efficient and scalable estimation of at least F(£) for a large number of qubits.

[0024] The main roadblock for a scalable URB stems from purity being a quadratic function of a quantum state, in general requiring knowledge of all its components, e.g., as described in

[0012] requiring state tomography within a RB protocol. The efficient estimation of purity, however, was one of the first problems giving rise to the Randomized Measurements (RM) and shadow tomography techniques [30, 33, 34]. While for the particular case of purity, the number of required measurements for a given accuracy still scales exponentially, the exponent for RMs is significantly smaller than for full tomography

[0030] ,

[0025] In [35, 36] is has been shown that where here U =® "=1Utis a tensor product of local random unitaries Ubwith EU£~Haardenoting averaging with the Haar measure over each local U( acting on a single qubit q,. s and s’ are n- bit strings obtained by measuring in the computational basis, i. e.,. is >= >, s = (s1(...,sn), h(s,s') is the Hamming distance (number of distinct bits) between them, and the expressions are probabilities of observing the n-bit string s upon randomizing with the unitary Z7and which are estimated in the experiment. Since Eq. (5) involves two copies of U and t / +, it suffices that the local unitaries belong to a unitary 2-design

[0037] , such as the uniformly- distributed Clifford group.

[0026] By a RM, here we will mean a random unitary operation U, followed by a projective measurement in the computational basis, l / |s). The concept of RM may be generalized to global unitary operations and / or Random Positive Operator Valued Measurements (POVM), as in

[0038] and equations similar to Eq. (5) above may be derived. In Eq. (5), this can equivalently be read as randomizing the state p with l / +, and then performing a computational basis measurement. Thus, the quantum processing device of the method according to the present invention comprises means for applying measurements to the n qubits. In particular, the means may be operative to implement a POVM or a projective measurement. As the quantum processing device also comprises means for applying a set G of quantum gates (and thereby is operative to apply random unitaries constructed from the gate set G) to the n qubits, the quantum processing device is operative to implement a randomized measurement.

[0027] Purity can alternatively be estimated through the so-called shadow of the state, constructed via RMs. That is, one can equivalently construct proxy states psthrough the probabilities PyS)and compute the purity through products tr[psps,]

[0029] , This is an equivalent approach with similar performance guarantees

[0030] ; Eq. (4) may be used within a RB-like protocol in a straightforward way.

[0028] Thus, in one example, calculating of the estimator values of the purity may be on the basis of estimator values of the outcome probabilities P® (see Eq. (5)). Operationally, the purity is really truly important once in light of an associated fidelity

[0039] : notice that the probabilities can be “recycled” in a straightforward way to estimate the fidelity of p with respect to an ideal pure state \ip}, through a slight modification to Eq. (4), as where here now Q®: = |(ij&|l / |s)|2is the probability of observing the n-bit string s upon randomizing with the unitary U, in particular with U =®”=1Utand measuring \ip). This comes at the expense of estimating the ideal probabilities but nevertheless this can be done classically in an efficient way, at least when the Utcan be replaced by single-qubit Clifford unitaries.

[0029] Within this application, we distinguish estimators of statistical parameters by a circumflex, e.g., 1 may represents the estimator value of a random variable X.

[0030] Due to the above, classical post-processing of the outcomes of the randomized measurements allows to calculate an estimator value of an average purity of all output states pr.. of the circuits of depth m. Classical post-processing is performed by a classical computer in one example.

[0031] From the above, it follows that the unitarity of a noisy quantum circuit should be a function of the circuit depth m, and in particular, the unitarity should decrease with the number m of layers of the quantum circuit Cm. Thus, the method according to the present invention further comprises conceiving a fitting model for a data set {(m, $m)} of the estimator values of the purity and the circuit depth m, wherein the unitarity u of the noisy gates of the randomly generated circuits is a parameter of said model, and fiting the model to the data set {(m, $m)} to thereby obtain an estimation of the value of the unitarity «. The unitarity u may be understood as the average fraction of the coherence of a channel describing an average layer noise.

[0032] In one embodiment of the method according to the present invention, said measurement means may be operative to measure in the computational basis |s) = |slr...,sn), = 0,1, of the n qubits, and said method may further comprise generating a plurality of Nw randomization circuits W, each comprising a randomly sampled randomization instruction with gates of the gate set G, and wherein providing the randomized measurement outputs for the output state comprises for each randomization circuit W applying said randomization circuit to said output state pcfollowed by applying said measurement means to the n qubits to thereby measure in the computational basis to obtain a measurement outcome s = (s1(...,sn). The method according to the embodiment includes RM as discloses, e.g. [35, 36] and mentioned above. In this case, the measurement outcome is a bit-string which is particularly well-suited for classical postprocessing as shown below. This embodiment allows to estimate the outcome probabilities These probabilities may be used for the estimation of the average purity as explained below. Due to the statistical nature of quantum mechanics, each randomization circuit W must be applied a plurality of Nmeas times to the same output state

[0033] In one embodiment, the method may be such that for each circuit depth m the average purity of all output states pCmvof the circuits Cmof depth m is estimated on the basis of the estimator values P^CmiVthe outcome probabilitiesPwpm,v = tr +] of the randomized measurements according to wherein h(s, s') is the Hamming distance between the measurement outcomes s = (st, ... , sn) and s' = (s / , ..., snr), Nvis the number of distinct random initial states pvfor the circuit Cm, Nwis the number of distinct randomization circuits, and NCmis the number of circuits Cmwith depth m. When there is exactly one initial state associated to each quantum circuit Q, as mentioned above, Nv= 1. The quantity may be calculated on a classical computer, for example by parallel processing. As explained above, P^CmiVis the estimator value of the outcome probability

[0034] As explained above, the unitarity only quantifies how unitary gnoisyis, but not whether it is the target unitary, i.e. , it does not measure whether gnoisyis far from the ideal g. In Eq. (2), the average gate fidelity was introduced as a figure of merit aiming to quantify this distinction. In one embodiment of the method according to the present invention the average gate fidelity may be estimated via an additional postprocessing step. To this end, one embodiment of the method according to the present invention further comprises: estimating, by a simulation on a classical computer, for each quantum circuit Cma value of outcome probabilities of a noiseless randomized measurement applied to a noiseless output state obtained by a noiseless application of the quantum circuit Cmto a noiseless random initial state pv; calculating, by the classical computer, for each circuit depth m, an estimator value of an average fidelity of circuits of depth m on the basis of the measurement outcomes of the quantum processor and the classically calculated noiseless outcome probabilities; conceiving a fitting model for a data set {(m, §m)} of the estimator values of the average fidelity of circuits of depth m and the circuit depth m, wherein an average gate fidelity is a parameter of said model; fiting the model to the data set to thereby obtain an estimation of the value of the average gate fidelity.

[0035] This embodiment is particularly useful when the quantum circuits Cmare classically simulable. The average fidelity may be understood as an average fidelity of the circuits Cmwith circuit depth m. The average gate fidelity may be understood as the average fidelity of a single layer of a quantum circuit comprising a plurality of layers.

[0036] In one embodiment, calculating the estimator value of the average fidelity of circuits of depth m may comprise determining, by classical simulation, the probabilities - tr[|s)(s|WCmpKcXw+] of the noiseless measurement outcomes and calculating the average fidelity wherein h(s, s’) is the Hamming distance between the measurement outcomes s = (%, ...,sn) and s' = Nvis the number of distinct random initial states pvfor the circuit Cm, Nwis the number of distinct randomization circuits, and NCmis the number of circuits Cmwith depth m. When there is exactly one initial state associated to each quantum circuit G® as mentioned above, Nv= 1. The quantity may be calculated on a classical computer, e.g., by use of parallel processing.

[0037] In one embodiment, the fitting model for the average purity is of the form Aum+ ^ = wherein 0 ≤ A ≤ 1 is a constant. Here, u is the average unitary of the noisy gates defined above. Such a model is particularly well-suited for an embodiment where the generated quantum circuits form an e-approximate unitary 2-design (see below) or a unitary 2-design, as one may show analytically in this case (see below) that the purity follows the above- mentioned exponential decay. In other cases, the above fiting model for the average purity may also be used, but the unitary u estimated by fitting the model to the data set may be a less accurate result for the true value of the unitary.

[0038] In one embodiment, the fitting model for the average fidelity of circuits of depth m may be of the form 2fm= afm+ — , where 0 ≤ a ≤ 1 , and a polarization is defined in terms of the average gate fidelity F of an average layer noise with respect to the identity averaged over the Haar measure, F wherein £ is the average layer noise. As for the purity, such a model is particularly well-suited for an embodiment where the generated quantum circuits C^form an e-approximate unitary 2-design (see below) or a unitary 2-design, as one may show analytically in this case (see below) that the average fidelity for the circuits of depth m indeed follows the above-mentioned exponential decay.

[0039] Thus, in one expedient embodiment of the method according to the present invention, generating the quantum circuits Cm is such that the quantum circuits Gaform an e-approximate unitary 2-design. In one example, the quantum circuits Cm are generated such that they form a unitary 2-design. Then, the average composition of the noisy layers is approximately equivalent to a depolarizing channel. In this case, the average purity and the average fidelity both follow an exponential decay as mentioned above.

[0040] In one embodiment of the present invention, the set G of quantum gates comprises a first set Gi of single-qubit gates and a second set Ga of two-qubit gates. Such a gate set G is native to many available quantum processing devices. In one embodiment, the set G consists of the first set Gi and the second set Gz.

[0041] In a further expedient embodiment, each of the m sets of instructions comprises a first instruction 1$ with gates of the first set G 1 performed in parallel, and a second instruction with gates of the second gate set G2 performed in parallel, wherein the first instructions Z® are sampled according to a first probability distribution Qi, and the second instructions Z® are sampled according to a second probability distribution Q2, and the first and second instructions are subsequent instructions. In one embodiment, each of the m sets of instructions consists of the first and second instruction. In particular, each set Lmof instructions is of the form Lm= One maVsaY thatthelayer A® >s a composite layer. Then, the quantum circuit C,may be represented as Cm= ... L^ with layers Lmas defined before.

[0042] Such circuits Cmare also denoted as “Q-distributed random circuit with a depth m” (see Ref.

[0025] ). In one embodiment, the set Gi of single-qubit gates may be a unitary-2 design. In an expedient embodiment, the set Gi of single-qubit gates may comprise the set of single-qubit 'Clifford gates. In a preferable example of this embodiment, the first probability distribution Qi is a uniform distribution. When Gi is the set of single-qubit Clifford gates and the first probability distribution Qi is the uniform distribution, a unitary 2-design is separately generated on each qubit. Then, it is preferably that the set G2of two-qubit gates and the second probability distribution Q2are such that the random Q -distributed circuits are highly scrambling, as defined in [25, 26], which can be achieved by it containing at least an entangling gate with high probability. While this does not ensure that the quantum circuits Cmapproximate a unitary 2- design in an increasing number of qubits, it suffices in practice for mid-scale systems. This is discussed in detail below.

[0043] In one embodiment, the method further comprises generating, for each quantum circuit Cnita state preparation circuit VCmcomprising a randomly sampled state preparation instruction with gates of the gate set G, and preparing the random initial state pvfor said circuit Cmby applying said state preparation circuit VCmto a predetermined quantum state p0which is the same for all quantum circuits. In one example, the predetermined quantum state is the state where each of the n qubits is in the logical state |0>, i.e., p0= |0 ...0)(0 ...0|. Such a state may be reliably (i.e. noise-free) prepared with state-of-the-art quantum processing devices. In the above embodiment, the initial state is random in that there is a different initial state for each quantum circuit and the initial states being randomly sampled according to a distribution. In this case, one may also say that the state preparation circuit is prepended to the quantum circuit Cm, i.e., for each quantum circuit Cm, the circuit is applied to the predetermined quantum state p0.

[0044] In one example of the above embodiment, each state preparation instruction V may comprise instructions with gates of the first set Gi performed in parallel and is sampled according to the first probability distribution Qi. I.e., the state preparation instruction comprises the instruction to apply a single-qubit gate to each qubit, and the single-qubit gates are randomly sampled. In one example, the state preparation instruction consists of the instruction to apply a single-qubit gate to each qubit

[0045] In one embodiment, each randomization instruction W may comprise instructions with gates of the first set Gi performed in parallel and may be sampled according to the first probability distribution Qi. Le., the randomization instruction comprises the instruction to apply a single-qubit gate to each qubit, and the single-qubit gates are randomly sampled. In one example, the randomization instruction consists of the instruction to apply a single-qubit gate to each qubit

[0046] According to a second aspect of the present invention, there is provided a computing system, said computing system comprising a classical computer and a quantum processing device, said quantum processing device including a plurality of n 2 qubits, ≥ means for applying a set G of quantum gates and means for applying measurements to the n qubits; wherein said classical computer is operative to generate, for each of a plurality of different circuit depths m > 1 , a plurality of quantum circuits Cra, each comprising m sets of randomly sampled instructions with gates of said set G of quantum gates; wherein said quantum processing device is operative to apply, for each quantum circuit Caof circuit depth m, said quantum circuit Cmto a random initial state pvof the n qubits to thereby obtain an output state prthereof, and to apply said measurement means to the n qubits so as to provide an outcome of a randomized measurement of said output state; wherein said classical computer is further operative to calculate, by classical postprocessing of the measurement outcomes, an estimator value of an average purity of all output states pCm Vof the circuits of depth m on the basis of the measurement outcomes of the randomized measurements and to fit a fitting model for a data set {(m, $TO)} of the estimator values of the purity and the circuit depth m to the data set, wherein a unitarity it of the noisy gates of the randomly generated circuits G„ is a parameter of said model, said unitary being indicative of the average contribution of the coherent noise, to thereby obtain an estimation of the value of the unitarity u.

[0047] The quantum computing system according to the second aspect of the present invention is operative to implement the method according to the first aspect of the present invention. Everything that was said above in relation to the method also applies to the quantum computing system. The classical computer may comprise means for inputting the fitting model. The fitting model may be conceived by a user of the method. The fitting model may be as described above in relation to the method, but it is not limited to this.

[0048] In the following description, the invention will be specified in greater detail by way of examples, with reference to the drawings:

[0049] Figure 1a is a schematic representation of an ideal quantum computation on n qubits, Figure 1b is a schematic representation of a more realistic, noisy quantum computation on the n qubits,

[0050] Figure 1c schematically illustrates the increasing impurity of a qubit state with an increasing number of applied noisy unitary operations,

[0051] Figure 2a schematically illustrates an application of an Q-distributed circuit G?to an initial state through a sequence of m random layers Li made up of only parallel layers of single- and two- qubit gates,

[0052] Figure 2b schematically illustrates a sequence of quantum operations applied in an embodiment of the method according to the present invention, said sequence including a layer of random single-qubit Clifford gates V prepended to the circuit Cm depicted in Figure 2a, defining a circuit Q, and a final measurement which is randomized with a different layer of random single-qubit Clifford gates W,

[0053] Figure 3 is a diagram which allows to estimate whether average noise is close to Pauli noise for a system of n = 10 qubits,

[0054] Figure 4 schematically illustrates an example of a layout of qubits of a quantum processing device with n = 20 qubits,

[0055] Figure 5 is a diagram that illustrates the results of a simulation of the decay of purity with the circuit layer depth for systems with n = 2, 4, 6, 8 or 10 qubits,

[0056] Figure 6a is a diagram that illustrates the results of a simulation of the decay of the fidelity with the circuit layer depth for systems with n = 2, 4, 6, 8 or 10 qubits,

[0057] Figure 6b is a plot that displays all extracted average layer fidelities and unitarities for an increasing number of qubits, together with ranges of values for potential Pauli unitarities according to equation (17),

[0058] Figure 7 is a comparison between results for the estimation of the layer (gate) fidelity obtained with (a) the method according to the present invention (b) with Mirror Randomized Benchmarking (MRB).

[0059] Brief description of the drawings:

[0060] Figure 1: (a) An ideal quantum computation on n qubits can be described by an initial state |0>®n, a sequence of unitary operationsrU.i, and a computational basis measurement with output s G {0,1}”. More realistically, (b), the initial state can be statistically mixed and correlated among qubits, and operations are subject to noise due to interactions with external degrees of freedom | et> that further reduce the state purity. The increasing impurity of the state Pt after the ishoperation can be depicted as a shrinkage of the 2n-1-dimensional Bloch vector, |v01 ≥ |vx | ••• ≥ |vm|; ≥ such shrinking can be associated on average to the unitarity or coherence amount of the noise, which is a resource intensive quantity to characterize experimentally as n increases.

[0061] Figure 2: fl-distributed circuits and RMs: In (a) an fl-distributed circuit Cmis applied on an initial state |0)®" through a sequence of random layers Li, L2,..., Lm~ fl made up of only parallel (according to device topology) layers of single- and two-qubit gates, Gi and G2, respectively, and a computational basis measurement is done after the mthlayer. In (b), a layer of random single-qubit Clifford gates V is prepended to the circuit G„, defining a circuit C^, and the final measurement is randomized with a different layer of random single-qubit Clifford gates W, generating a n-bit string s; a set of such measurements can be processed according to Protocol 4.2 to estimate the average unitarity of the corresponding fl-distributed circuits.

[0062] Figure 3: Pauli unitarity region. Case n=10; the dashed blue region denotes all average unitarity and fidelity combinations that are not possible for Markovian CP noise, the green shaded region denotes values within the bound for unitarity of Pauli noise (17), and the white region denotes all values outside the bound, guaranteed to have coherent error contributions.

[0063] Figure 4: Backend Topology: We consider a backend with 20 possible qubits (circles), with each qubit labeled with a given integer and connected to another as depicted by the corresponding edges. The fl -distributed circuits we consider will employ single qubit gates in any qubit being employed and two-qubits gates between any pair of connected qubits.

[0064] Figure 5: Purity decay simulation. Simulation of purity decays computed according to Protocol 4.2 for fl -distributed circuits using the gate set G = ClifxU {CZ}, with Clift being the uniformly-distributed single-qubit Clifford group. Individual plots correspond to a given number of qubits n (taken in increasing number from the layout in Fig. 4), whereby smaller points correspond to Median of Means (MoMs) estimators of purity with Eq. (11 ) at a given depth m, bigger points correspond to the average purity, violins show the distribution of the MoMs estimators, and lines correspond to a least-squares fit of the averages to the decay model in Eq. (13), whereby the average unitarities u in corresponding number of qubits n are estimated. Sample parameters: Nc^ = 12, Nw= 9, WOTeas= 210, K = 9 MoMs.

[0065] Figure 6: Purity decay simulation and average layer fidelities and unitarities, (a) Simulation of fidelity decays according to Protocol 4.2 corresponding to the same fl-distributed circuits and parameters giving probabilities with purities displayed in Fig. 5; here similarly individual plots correspond to a given number of qubits n, small points correspond to MoMs estimators of Eq. (12) at the given depth, larger points correspond to the respective average purity, and lines correspond to least-squares fit of the averages to the model in Eq. (15), whereby the corresponding average layer fidelity F in the respective number of qubits n is extracted according to Eq. (16). Sample parameters: Nc^ = 12, Nw= 9, Nmeas= 210, K = 9 MoMs. In (b) a plot displaying all extracted average layer fidelities and unitarities in increasing number of qubits is shown, together with ranges of values for potential Pauli unitarities according to Ineq (17).

[0066] Figure 7: Comparison with MRB: Fidelity decays for Q-distributed circuits considering a uniformly distributed single-qubit Clifford group and a CZ gate on a line of 9 qubits with nearest- neighbour connectivity. In both figures, individual plots correspond to different number of qubits, small points represent individual state fidelity outputs at a given depth (for (a) taken over Median of Means (MoMs) estimators), bigger points represent average state fidelities at corresponding depth, and lines correspond to least-square fits of the corresponding averages to the exponential decay model Eq. (12). In (a) we show the outputs with our technique using sample parameters Nc^ = 12, Nw= 9, Nmeas= 210, K = 9 MoMs, while in (b) we show the outputs of MRB using 20 circuit samples with 20 set of Pauli layer samples each.

[0067] The complexity of noise mechanisms limiting the performance of quantum computing makes it crucial to be able to efficiently estimate its leading error sources. In particular, quantifying whether the instructions on a quantum processor are in practice close to being unitary has important consequences: however, this is generally a resource-intensive task in system size. Building upon developments for scalable benchmarking and randomized measurement techniques, we develop a protocol allowing to estimate the average unitarity of operations in a digital quantum device in a way that is efficient and feasible for intermediate-size quantum systems. We demonstrate our technique through numerical simulations with up to 10 qubits and discuss both the limitations and prospects for extending our technique to arbitrary scaling.

[0068] 1 Introduction

[0069] The main obstacle for quantum computing undeniably remains the presence of noise, caus- ing a multitude of errors that limit the usefulness of currently-available quantum processors. As not all quantum noise is made equal, figures of merit such as the fidelity of a quantum operation are by definition insensitive to many of the details of such noise.

[0070] Quantum noise can be described as an undesirable quantum transformation mapping the ideal noiseless output state of a quantum computation into some other valid, albeit unexpected, output quantum state. Once assuming that such noise is Markovian, i.e., it only depends on the input state and no other external context variables, a comprehen- sive classification of all possible errors can be made [1], but in particular it is extremely important to be able to distinguish between coherent and incoherent error contributions.

[0071] Coherent noise can be described as a deterministic, undesired unitary transformation, e.g., it can arise due to imperfect control or calibration of quantum gates, and also due to certain types of crosstalk [2] . Incoherent or stochastic noise, on the other hand, is described by iion-unitary transformations and arise purely statistically, e.g., a bit-flip occurring with a certain non-zero probability. The reasons why distinguishing between these is important, include: i) Coherent errors can, for example, be effectively suppressed by good experimental quantum control, while incoherent errors in isolation can be efficiently mitigated [ - 7] and fully corrected [ , ]. ii) Coherent errors can accumulate in a much more detrimental way and provide error- rate estimates, such as average gate- fidelity, that can differ by orders of magnitude from its worst-case counterparts (i.e., fault-tolerance thresholds) [ - ].

[0072] Hi) The interpretation of average gate fidelity as a reliable and meaningful error rate depends heavily on the noise having a low coherent contribution [ , - ].

[0073] The average coherence of a noisy quantum gate can be understood as a measurement of the rate at which noise shrinks the n-dimensional Bloch ball, as depicted in Fig. 1, and can be measured by its unitarity, which in turn can be estimated operationally by Unitarity Randomized Benchmarking (URB) [ ]. Despite being efficient and robust to State Preparation and Measurement (SPAM) errors, URB is not scalable in system size, as it requires estimating an exponential amount of expectation values within an Randomized Benchmarking (RB)-like protocol1. The reason for this is not URB per se, but rather that fundamentally, the unitarity is a second-order functional of the noise, akin to the case of the so-called purity with respect to a quantum state.

[0074] Figure 1: Coherence of noise and loss of purity: (a) An ideal quantum computation on n qubits can be described by an initial state 10)®", a sequence of unitary operations and a computational basis measurement with output s 6 {0, 1}”. More realistically, (b), the initial state can be statistically mixed and correlated among qubits, and operations are subject to noise due to interactions with external degrees of freedom |e^) that further reduce the state purity. The increasing impurity of the state p,;after the ithoperation can be depicted as a shrinkage of the 2n”1-dimensional Bloch vector, |VQ | > )vil ≥ . . . |v ≥m|; such shrinking can be associated on average to the unitarity or coherence amount of the noise, which is a resource intensive quantity to characterize experimentally as n increases.

[0075] In general, the complexity of characterizing quantum noise in a comprehensive way typically scales exponentially with system size [ , ]. Even most efficient protocols estimating a single figure of merit, such as those within the family of RB [ ], fail to do so

[0076] 1So-called Speckle Purity Benchmarking is a technique that can indirectly estimate average unitarity for larger systems: however it is not a standalone technique but a byproduct of Cross- Entropy Benchmark- ing [ ] , which itself is famous for precisely requiring classical computation scaling exponentially in system size, and serving rather as a fidelity proxy under certain conditions [ , ]. at scale. However, with the quantum industry inevitably moving towards larger systems, there is a pressing need to find ways to generalize such tools to larger systems.

[0077] Recently scalable RB-based techniques such as Mirror Randomized Benchmarking (MRB) and Binary Randomized Benchmarking (BiRB) [ ]. among others | , ]. have been developed, which are able to estimate the average fidelity of sets of quantum op- erations in a scalable, efficient and SPAM-robust way, for a. large class of gate sets. On the other hand, Randomized Measurements (RM) techniques, and so-called classical shadows, have enabled the efficient estimation of properties of many-body quantuni systems [ . ]

[0078] (including state fidelity and purity), linear properties of gate sets [ ], and characterization of processes with memory

[0032] ,

[0079] Here, we harness the results of [ - ]. together with a scalable RB-inspired protocol, to develop a standalone technique allowing to simultaneously estimate the average fidelity and average unitarity of layers of operations in rnid-scale quantum systems. We further identify an interval for the unitarity of so-called stochastic Pauli noise in terms of average fidelity, allowing to certify that the noise has a low coherent contribution.

[0080] In § 2 we introduce the main technical background, while in § 3 we summarize the main results regarding RMs. In § 4 we present our protocol and results, while in § 5 we demonstrate it with numerical simulations4. In the remaining § we discuss the bottlenecks for a scalable estimation of average coherence of noise, and finally we draw some conclusions and prospects to overcome mid-scalability in § 7.

[0081] 2 Average coherence of gates and its relation with purity and fidelity

[0082] It is well known that no physical system can be perfectly isolated. In particular, the loss of coherence of quantum states is one of the fundamental practical problems facing any quantum technology. In general, the dissipation of information to an external environment results in increased subjective uncertainty as to what the quantum state of a system is, which can be quantified by the purity of the respective density matrix p, given by The purity takes extremal values of 1 for a pure state, and 2""' for a maximally mixed state, and it can also be written as the Schatten 2-norm ||p|||, where ||X|(2 = (tr(XX’f)]1 / 2.

[0083] Logical operations or gates Q in a quantum computer are described by unitary oper- ators, which by definition preserve the purity of quantum states, i.e., for G(pm) = pout, we have tr(p2n) The corresponding, real noisy operation, ( / noisy; however, is de- scribed by a more general Completely Positive (CP) map, such that for ^noisy (din) = Pout; we have tr[PoUt] tr[ / 4i] - The reduction in purity of states when acted on by a noisy channel is related to the unitarity of such channel, a measure of “how / ar” lAoisy is from being unitary. Because locally we can always relate a noisy channel with an ideal gate by f / noisy := E o G ([1]) for some CP map , the unitarity of the noisy gate equivalently measures how far E is from being unitary.

[0084] The average unitarity of a quantum channel can be defined as in | |, by where A(-) :— £( ■ — l / 2n), with 1 being an n-qubit identity operator, and the normaliza- tion 2n / (2n— 1) ensuring that 0 ≤ u(£) ≤ 1, and with u(£) = 1 if and only if E — and

[0085] Experimental execution on IQM’s superconducting hardware is currently being organized. hence the noisy operation ( / noisy — is unitary. This definition in Eq. (1) thus implies that the average purity of a noisy state is a combination of average unitarity and trace-decrease and / or non-unital ’■ terms.

[0086] The unitarity only quantifies how unitary ( / noisy is, but not whether it is the target unitary, i.e., it does not measure whether Pnoisy is far from the ideal Q. A figure of merit aiming to quantify this distinction is the average gate fidelity, where implicitly we denote F(£) := F(8,T). the average gate fidelity of 8 with respect to the identity I (and in a slight abuse of notation, = I' / ’X / ’I)- The average gate fidelity satisfies 1 / (2” — 1) < F(£) < 1, with F(8) = 1 if and only if 8 = T (equivalently if and only if £noisy = G'Y With respect of F, the average unitarity satisfies as shown in

[0012] , with saturation occurring for a depolarizing (purely incoherent) channel.

[0087] The unitarity can also be written in terms of fidelity as u(£) , where is the adjoint map of 8'*. Thus, the average unitarity can also be understood as a second-order functional of the fidelity of 8 quantifying on average how distinguish- able is the composition 8'^8' from the identity. Under certain assumptions ' F(£) can be estimated through RB, and the unitarity u(£) can be estimated through URB |. Both techniques are not scalable in number of qubits, however, recently the MRB and BiRB variants were developed in lowing efficient and scalable estimation of at least F(£) for a large number of qubits.

[0088] 3 Purity and fidelity via randomized measurements

[0089] The main roadblock for a scalable URB stems from purity being a quadratic function of a quantum state, in general requiring knowledge of all its components, e.g., as described in [ quiring state tomography within a RB protocol. The efficient, estimation of purity, however, was one of the first, problems giving rise to the RM and shadow tomography techniques [ , , |. While for the particular case of purity, the number of required measurements for a given accuracy still scales exponentially, the exponent for RMs is significantly smaller than for full tomography [ |.

[0090] We will focus on the result

[0091] 3A CP map 8 such that trfo(p)] < tr(p) is trace-decreasing, and one such that f(l) 1 is non-unital.

[0092] 4The adjoint map of a CP map # is defined by tr[A$(B) = tr[W(A)jBj, or equivalently by conju- gating the Kraus operators of <h.

[0093] 5That E approximately models the average noise of a gate set {&} in a temporally-uncorrelated (Marko- vian) . time-independent and gate-independent way. where here U = ®"=1Uj is a random unitary, with Eu^Haar denoting averaging with the Haar measure over the each U^, s and s' are mbit strings, h(s, s') is the Hamming distance (number of distinct bits) between them, and the (5) are probabilities of observing the n-bit string s upon randomizing with U, and which are estimated in experiment. Since Eq. (4) involves two copies of U and Ufi it suffices that the local unitaries Uj belong to a unitary 2-design [ ], such as the uniformly-distributed Clifford group.

[0094] By a RM, here we will mean a random unitary operation U, followed by a projective measurement in the computational basis, U|s) ”, In Eq. (5), this can equivalently be read as randomizing the state p with V = U^, and then performing a computational basis measurement.

[0095] Purity can alternatively be estimated through the so-called shadow of the state, con- structed via RMs. That is, one can equivalently construct proxy states psthrough the probabilities Pu(s) and compute the purity through products tr(psps / ) [ ]. This is an equivalent approach with similar performance guarantees [ ]; here we employ Eq. (4) because it can be used within a RB-like protocol in a straightforward way.

[0096] Operationally, the purity is really truly important once in light of an associated fi- delity [ ]: notice that the probabilities can be recycled in a straightforward way to estimate the fidelity of Q with respect to an ideal pure state 1-0), through a slight modifi- cation to Eq. (4), as where here now is the probability of observing the n-bit string s upon randomizing with U = Uj and measuring This comes at the expense of estimating (s) the ideal probabilities Qy , but nevertheless this can be done classically in a efficient way. since the Uj can be replaced by single-qubit Clifford unitaries.

[0097] 4 Average unitarity by randomized measurement correlations

[0098] 4.1 Setup and Notation

[0099] While we do not require the exact same setup as in MRB or BiRB, we mainly follow the framework and notation of [ ]. We will, in particular, only consider single-qubit and two-qubit gate sets, Gi and G2, distributed according to probability distributions £2i and ^2, respectively. We will refer to instructions with these gate sets on n-qubits, simply as layers, and denote the corresponding set of layers by E(G), where G ~ Gi U 62- In practice, a rule on how to apply the sampled gates must also be specified, depending e.g., on the topology, connectivity, and the desired density of the gates.

[0100] We construct n-qubit random circuits, of layer circuit circuit depth m, of the form (7)

[0101] 6More strictly, this corresponds to a RM element, and the concept of a RM can be formalized with Random Positive Operator Valued Measurement (POVM)s, as in

[0038] where the right-hand side is read (or acts) from right to left, and where each is a composite layer, made up of subsequent applications of a layer G IL(Gi), of only parallel single-qubit gates, and a layer Q ' € ILfGa), of only parallel two-qubit gates. We will say that all L, G L(G) are random layers distributed according to fl, which is such that refer to Cmin Eq. (7) as a Q,- distributed random circu

[0102] Figure 2: fl-distributed circuits and RMs: In (a), an fl-distributed circuit Cmis applied on an initial state |0)®" through a sequence of random layers Li, L2, . . . , Lm, ~ fl made up of only parallel (according to device topology) layers of single- and two-qubit gates, Gj and G2, respectively, and a computational basis measurement is done after the mthlayer. In (b), a layer of random single-qubit Clifford gates V is preppended to the circuit CTO, defining a circuit C*vand the final measurement is randomized with a different layer of random single-qubit Clifford gates W, generating a n-bit string s; a set of such measurements can be processed according to Protocol to estimate the average unitarity of the corresponding fl-distributed circuits.

[0103] We use the term depth to mean implicitly the number of layers entering the given fl- distributed circuit, before transpilation to a given basis gate set. These concepts, together with that of RMs, are illustrated in Fig. 2. We also point out that while the choice of the pair (G, fl) is a priori arbitrary, here we will only consider Gi Clif i being the Clif- ford group (or effectively any other efficient single-qubit unitary 2-design) with a uniform distribution fli .

[0104] Finally, we distinguish estimators of statistical parameters by a circumflex, e.g., X might represent the estimator of a random variable X. 4.2 The protocol

[0105] With this setup, the protocol consists of:

[0106] 1. Preparation-. Generate Acy, samples of n-qubit ^-distributed circuits Cmfor all given circuit depths m, each with a different layer V E L(Clif i) sampled uniformly, preppended to it. We will denote (8)

[0107] 2. Quantum Execution: Estimate the probabilities (9) of observing the n-bit string s, for all circuit depths m, all circuit samples and a given number Nyj of layer samples W E IL(Gi), sampled according to Qi.

[0108] We denote the corresponding estimator quantities, given a finite number of measure- ments Ameasby where Xj is a random variable describing the ithprojective measurement of the n- qubits in the computational basis, with ls(x) = 1 if x = s or 0 otherwise.

[0109] 3. Classical post-processing: Estimate the average purity tr(p^) for all depth rn states 0rn. ■= CnWG0X0|)VtCL via where h(s, s) is the Hamming distance (i.e., number of distinct elements) between the n-bit strings s and s'.

[0110] Optionally, and if the Q-distributed circuits are efficiently classically simulable, es- timate the probabilities QwV*:= |(siW^idGb’md^| for all noiseless states. of depth rn — with the label (id) here denoting noiseless quantities — , and the corresponding average fidelities via

[0111] Both estimators can be rendered unbiased and improved by certain considerations as explained

[0112] Whenever the Q-distributed circuits are generated by layers containing non-Clifford gates in a way that becomes impractical to simulate (e.g., whereby there is a high-density of such non-Cliffords or for a large number of qubits), the average layer fidelity can alter- natively be estimated through MRB [ ] ; the trade off is having to construct and measure corresponding mirror circuits. The estimations of Protocol and MRB by definition will agree upon employing the same ensemble of layers; in Appendix F we observe numerically such agreement. 4.3 Main Results

[0113] Result 1 (Exponential decay). Throughout Appendices wc show that, under the circumstances detailed below,

[0114] Etr where 0 ≤ A ≤ 1, and n(£)PPExl(14)is the average layer unitarity of the noise, where 8 := A,;TOi5y£j is approximately the noise channel corresponding to any layer L, with associated map £.((•) := Lj (-)LJ , and F(<fif£) is the average fidelity of 8^8 with respect to the identity; this is an equivalent expression to that obtained originally in I

[0115] Similarly, the decay in average fidelity can be seen to correspond to where 0 ≤ a ≤ 1, and (16) similar to a usual RB decay for unital SPAM.

[0116] In such case, the estimators in Eq. ( ) and Eq. ( ) of protocol can be fit to the respective decays in depth m, whereby both average layer unitarity and fidelity can be estimated.

[0117] Result 2 (Pauli unitarity interval). Pauli noise is a special case of incoherent noise de- scribed by a channel A(-) 52pePaulin« / ’ / ’(•)£, where ap is the Pauli error probability p PpP associated to a n-qubit Pauli P. Pauli noise is of high relevance to all error suppression, correction and fault tolerant quantum computation - , - ]. In Ap- pendix . we show that the unitarity of the Pauli-twirled channel 8° of 8. i.e., the Pauli channel with the same Pauli error probabilities as 8, is bounded by ≤ ≤ where / (£) is the polarization of 8, given by Eq. ( ), and r(8 ) := 1 — F(£) is the average infidelity of 8.

[0118] 4.3.1 Conditions for an exponential decay

[0119] While a general functional form in terms of any Markovian noise model for the average sequence purity and fidelity can be obtained, as in Appendic ct> . and , ensuring that it will follow a simple exponential decay as in Eq. (13) relies on the noise satisfying certain approximate conditions and the pair (G, O) having certain properties, similar to any RB- based technique. Aside from standard assumptions such as Markovianity (i.e., that, noise is not temporally correlated), time-independence and weak gate-dependence ' , Result 1 requires the following: rHere meaning with time- and gate-dependent contributions being negligible on average; see e.g.. [ ]. Ai. That noise is approximately trace-preserving and unital, i.e., without significant leak- age or entropy-decreasing contributions.

[0120] Ag. That the O-distributed circuits approximate a unitary 2-design, to the effect that — p)l / 2n, where p 1 o ≤nly dependent on X; i.e., the average composition of noisy layers is approximately equivalent to a depolarizing channel encoding some information about X. This is formalized and discussed in Appendix .

[0121] Assumpti< nplies that the purity decay only depends on the unitarity of the average noise, and not on its non-unital or trace-decreasing contribution. Otherwise, if this condition is not satisfied, the decay will be a convex combination of average unitarity and trace-decreasing contributions of the average noise.

[0122] Assumption implies that we can identify u(E) = p2as given by Eq. (14), i.e., the average unitarity is simply equal to the polarization of the average composition of noisy layers. This assumption is slightly stronger than that in [ - , ], and generally means that up to the second moment and a small positive e, the fl-distributed circuits we employ should reproduce the statistics of the uniform Haar measure on the unitary group on the n qubits. While the choice of (Gy ffy) being the uniform single-qubit Clifford group already generates separately a unitary 2-design on each qubit (projecting noise to a Pauli channel

[0051] ), the choice (62, ^2) should be such that the random fl-distributed circuits are highly scrambling, as defined in [ , J, which can be achieved by it containing at least an entangling gate with high probability. While this does not ensure that the circuits will approximate a unitary 2-design in an increasing number of qubits, it suffices in practice for mid-scale systems. This is discussed in detail in Appendix .

[0123] 4.3.2 Pauli noise unitarity

[0124] Incoherent noise is Trace Preserving (TP) and unital, and a distinction from coherent, noise is that it generates no net rotation of the state space [1]. Pauli stochastic noise is a particularly relevant kind of incoherent noise whereby all error contributions come from probabilities a.p of errors p PpP for all n-qubit Pauli operators5from occurring. Its relevance stems not only from that of the Pauli basis in quantum information theory and experiment, but also from its extensive application in quantum error characteriza- tion [ quantum error mitigation ] - ], quantum error correction [ , ] and in its connection with fault tolerance thresholds [ . , ] .

[0125] The Pauli-twirled noise channel of a given channel £, as defined in Result 2, can be efficiently approximated operationally by Randomized Compiling (RC) [ . ]. We relegate technical details to Appendix . but the reasons above imply that it is particularly relevant not only to minimize coherent contributions to noise but to at least have noise that has only Pauli contributions. Given an average layer (in)fidelity. the bound in Ineq. (17) allows to estimate whether average noise is close to Pauli through the average unitarity. Generally, for a small average infidelity, the bound will be tight and the average unitarity will need to approach the average polarization / (E); this can be seen more clearly for some fixed n as in Fig. .

[0126] The upper bound in Ineq, i ) overestimates the average unitarity of Pauli noise by a proportion of cross-products of Pauli error rates, and further assumes

[0127] 8Formal definitions can be seen in Appeir i

[0128]

[0129] Figure 3: Pauli unitarity region. Case n = 10; the dashed blue region denotes all average unitarity and fidelity combinations that are not possible for Markovian CP noise, the green shaded region denotes values within the bound for unitarity of Pauli noise ( 1, and the white region denotes all values outside the bound, guaranteed to have coherent error contributions.

[0130] TP noise, so while a unitarity outside the bound guarantees that average noise will contain coherent error contributions, the converse just points to average noise being close, to Pauli.

[0131] 4.4 Mid-scale

[0132] In § (5, we will discuss the reasons why Protocol is practical and feasible for at least 10-qubits and generally within tenths of qubits, which is what we refer to as mid-scale. This is in the sense that all classical aspects, i.e., steps 1 and can be managed with a standard current-technology laptop and with the quantum execution requiring a mild number of circuit. RM, and measurement shot samples, as exemplified in § 5.

[0133] 4.5 Unbiased and Median of Means estimators

[0134] There are at least two simple but important ways in which reliable results can be ensured by means of the estimators in Eq. ( ) and Eq. ( ): / ) using unbiased estimatorsuand it) using Median of Means (MoMs) estimators.

[0135] In

[0054] it is pointed out that, even though the probability estimator in Eq. (’ ) is faithful, i.e. E[P] — P (where we have dropped all indices), it is biased for any other positive integer power. A unique unbiased estimator can nevertheless be built for any power, and in particular, P2 — 1) is an unbiased estimator of P2. Thus all terms in Eq. ( ) and Eq. ( ) for the n-bit string s = s' should be computed by means of P2 in order to render average sequence fidelity and purity estimations unbiased.

[0136] 9An estimator of some statistical parameter is said to be unbiased or faithful when its expectation matches the real expected value of the parameter, or otherwise it is said to be biased. On the other hand, a simple but highly effective way to reduce the uncertainty asso- ciated to a mean estimator is to use MoMs estimators: suppose we have K samples of estimators , SrrP for the average fidelity of circuits of depth m in Eq. (12), then the MoMs estimator of such samples is := median(Mm • • • or similarly for the case of the average purity in Eq. ( ). In total, this requires circuit samples to be measured Armeastimes, which however. is much more robust compared with simply constructing a single mean estimator with the same amount of samples [ |. For- mally, this is ensured by concentration inequalities (under only, e.g., a finite variance) with a probability of observing outliers from the true mean decreasing exponentially in K [ ] .

[0137] 5 Simulation Results

[0138] We now demonstrate the execution of Protocol by means of simulation using Qiskit’s AerSimulator using a noise model as specified below.

[0139] We employ a simulator backed with native gate basis {r#, CZ, measurement, barrier}, where r@ is a single-qubit rotation around the X-axis with angle 9, CZ is a controlled-Z two-qubit gate, the measurement operator is a computational basis projective measurement operation and the barrier operation prevents operations from being transpiled together (this is important for separating logical layers once they are transpiled into the native gate basis). The backend considers 20 possible qubits with a topology (connectivity) as displayed in Fig. 4.

[0140] Figure 4: Backend Topology: We consider a backend with 20 possible qubits (circles), with each qubit labeled with a given integer and connected to another as depicted by the corresponding edges. The Q-distributed circuits we consider will employ single qubit gates in any qubit being employed and two-qubits gates between any pair of connected qubits.

[0141] The random Q-distributed circuits are generated by means of the edge-grab sampler, defined in | i, with a two-qubit gate density (the proportion of qubits occupied by two- qubit gates) of 1 / 4. This sampler considers layers with two-qubit gates only on connected qubits, and where a single logical layer consists of a parallel mixture of one- and two-qubit gates, i.e. there is no more than one gate acting on a given qubit in any layer.

[0142] All operations except barriers are modeled as noisy, with a set of parameters considering a range of relaxation times Tj, dephasing times T%, depolarizing parameters on both single- and two-qubit gates, readout errors, and gate duration parameters.

[0143] 10° io1102

[0144] Circuit Layer Depth (log)

[0145] Figure 5: Purity decay simulation. Simulation of purity decays computed according to Protocol for Q-distributed circuits using the gate set G = Cl if 1 U {CZ}, with Clif 1 being the uniformly-distributed single-qubit Clifford group. Individual plots correspond to a given number of qubits n (taken in increasing number from the layout in Fig. 4), whereby smaller points correspond to Median of Means (MoMs) estimators of purity with Eq. ( J i) at a given depth m, bigger points correspond to the average purity, violins show the distribution of the MoMs estimators, and lines correspond to a least-squares fit of the averages to the decay model in Eq. 1 ), whereby the average unitarities u in corresponding number of qubits n are estimated. Sample parameters: = 12, AM = 9. A'meas = 210and K = 9 MoMs.

[0146] With this setup, according to Protocol , we extracted decay rates of the average state fidelity and purity in increasing layer depths, and thus layer fidelity and unitarity, for Q-distributed circuits with up to 10 qubits. The set of qubits we considered ranged from qubit 1 to qubit 10, in increasing order, as displayed in the topology of Fig. 4.

[0147] In Fig. 5, we show the average purity decay in increasing depth for fl-distributed circuits using the gate set G = Clif i U {CZ}, with Clif] being the uniformly-distributed single-qubit Clifford group (i.e., fli assigning 1 / 24 probability for each Clifford and fig probability 1 of sampling CZ). We display only even qubit numbers n for easiness of visualization. The sample parameters we employ are Wc* = 12 random Q-distributed circuits, times Arw = 9 randomized measurement layers, and times fVmeas = 210shots per measurement; we furthermore employ unbiased estimators for square probabilities and K = 9 Median of Means (MoMs) estimators.

[0148] On the other hand, in Fig. (a), we used the same outcome probabilities, together with the simulated noiseless outcome probabilities, to compute the average state fidelity decay in increasing depth, and thus the corresponding average gate fidelities. We again show only even number of qubits for easiness of visualization. The noiseless probabilities were estimated with the same number of Arme3S- 210shots per measurement. Finally, Fig. (b) shows all average layer fidelities and unitarities in increasing number of qubits, together with the Pauli noise unitarity intervals as predicted by the bound in Ineq. I ).

[0149] Generally, while variances in the decays are relatively high, the averages show only a modest deviation up to 9 qubits. The noise model is dominated by stochastic errors, so it is expected that most points should fall within the Pauli unitarity bound; the cases of n = 6, 8 Gi — Clifford, G2 "Q2—{CZ; 1}

[0150] Figure 6: Purity decay simulation and average layer fidelities and unitarities, (a) Simulation of fidelity decays according to Protocol 4 2 corresponding to the same fl-distributed circuits and parameters giving probabilities with purities displayed in Fig. 5; here similarly individual plots correspond to a given number of qubits n, small points correspond to MoMs estimators of < ) at the given depth, larger points correspond to the respective average purity, and lines correspond to least-squares fit of the averages to the model in Eq. i ), whereby the corresponding average layer fidelity F in the respective number of qubits n is extracted according to Eq. (16). Sample parameters: = 12, #w= 9,

[0151] -■Vmeas = 210and K = 9 MoMs. In (b) a plot displaying all extracted average layer fidelities and unitarities in increasing number of qubits is shown, together with ranges of values for potential Pauli unitarities according to Ineq. ( ). are indeed outliers that would need to be investigated further in real experiment, in our case it most likely can be explained by both the qubit arrangement and the amplitude damping Ti parameters involved. It should be noted that for simulations in a common laptop, going beyond 10 qubits becomes quite computationally demanding. In Appendix show a few other examples considering other noise models, as well as a comparison with layer fidelity rates extracted by MRB, generally showing quantitative agreement within uncertainties inherent to both techniques

[0152] 6 Scaling Bottlenecks

[0153] 6.1 Sample Complexity

[0154] The main roadblock for estimating the average coherence of noise in large systems (or really even just the purity of quantum states) with RMs, is the required number of measurements to have an estimation within a given error, i.e., its sample complexity, which is then bydefinition manifested by the variance associated to such RMs. tatistical uncertainty stems from the sampling of the fl-distributed M elements, and the number of measurements per RM element; these to NQ* and Ayv in the estimators of Eq. ( ) and Eq. ( ) , and to pectively.

[0155] The main limitation to scalability is imposed by 2Vmeas: using local RMs via Eq. (4) to estimate purity to precision 1 / S / N\N, it was observed numerically in | | that it scales approximately as Nmeas~ 20<7onin number of qubits n. Generally, within the t l<i.->sical shadows framework, the required number of measurements can be seen to tially in a similar fraction of n and linearly in the actual purity of the state i

[0156] Mixedness reduces the sample complexity, as also observed numerically in | . md with

[0157] Q-distributed circuits, quantum states will naturally get increasingly mixed in increasing depth. Coincidentally, thus, a high percentage of coherent error contributes to a higher variance not only in average fidelity estimation but also in average unitarity estimation itself.

[0158] When it comes to fl-distributed circuit samples, when estimating average fidelity, the variance associated to it can be directly linked to the average amount of coherent noise

[0159] ]. How to control this uncertainty, as well as guarantees when choosing the number of sample circuits and number of measurements, has been widely studied for Clifford RB

[0160] ], furthermore. [ ] obtained a bound on the variance for the purity in circuits within (an impioved version of) Clifford URB, assuming unital noise. While we are not aware of analogous bounds on the variance for scalable RB (namely employing fl-distributed circuits), it has been observed numerically that similar behavior follows I at least when layers are made up only of either general, ors-of or Clifford gates themselw s} | |. and that Pauli tunhr y [ | gt neialh suppresses the variance of expectation values under any sequence of operations reg.it dless of whether noise is correlated (either spatially or temporally) or not [ |.

[0161] A prospect could therefore benchmark average Pauli-dressed gates, aiming to obtain a unitarity within Ineq. ith the trade-off being to have to sample an extra number of Pauli layers per circuit. Here we do not attempt to derive a rigorous bound on the variance for the average layer unitarity, but as we have demonstrated experimentally in § 5, while for a given total number of samples the uncertainty is larger for the unitarity than for fidelity, estimating it nevertheless remains tractable for mid-scale systems 6.2 Practical bottlenecks

[0162] There are two other practical considerations that have to be taken into account to estimate at which qubit scale Protocol 3 remains feasible: the SPAM factors in the exponential decay in Eq. ( ) (and / or Eq. ( )), and the number of terms entering in the sum of Eq. (11) (and / or Eq. (12)).

[0163] While our technique gives SPAM-independent estimates of the average unitarity, it can nevertheless make the exponential decays drop way too quickly, so that fitting becomes unfeasible. This is a shared problem of all RB-based techniques, and other than simply improving readouts and / or state preparation, there could be ways of removing the 2~raoffset, for example, but the multiplicative factor is still determined by the unitarity of the SPAM . More than a RB problem this might be an issue in general when trying to estimate global, high-weight properties on large systems, essentially because signal readouts eventually become too small.

[0164] On the other hand, while Eq. (4) and Eq. (5) are remarkable mathematically, in practice they involve a sum over all possible pairs of n-bit strings. Naively, this means summing up 4nprobability terms; for the case of the purity, this can be reduced at least to 2” , given that the Hamming distance is symmetric. This computation can be done easily on a classical machine in parallel, but it can also become rather unpractical (as it furthermore would need to be done for every single purity estimate). While this is a practical problem, it is fundamentally tied to the fact that the purity of a state involves all the elements of the density matrix.

[0165] 7 Conclusions

[0166] We have established a protocol enabling the estimation of the average coherence, or uni- tarity, of a broad class of quantum circuits in digital systems (independent of platform or architecture) with tenths of qubits. We have shown that reliable estimation of both operation coherence and fidelity in such systems can be done under mild conditions and with a reasonable sampling overhead. Additionally, we have obtained an upper bound for the unitarity of Pauli-twirled noise solely in terms of the average fidelity of the bare noise, increasing the importance of estimating the average unitarity. Our results have been inspired both by novel scalable Randomized Benchmarking (RB) techniques, as well as Randomized Measurements (RM) methods. We have demonstrated our results numeri- cally in simulations mimicking real hardware, considering up to 10 qubits.

[0167] A fully scalable estimation of the average coherence of noise will almost surely rely on modularity, i.e., somehow using smaller-scale unitarity information, which might involve e.g., smaller qubit subsets and / or with a simultaneous RB-like protocol, as recently done in [ ] to estimate holistic fidelities. A promising way of moving forward in this direc- tion is that of [ ], where the authors establish a method to estimate the global purity (or more generally entropy and entanglement) of a quantum state in polynomially-many measurements, relying on certain conditions essentially satisfied when spatial correlation lengths in the system remain finite. Their main result -phrased only for one-dimensional systems, i.e., with topology of a line-, implies that one can partition a (large) n-qubit

[0168] 10For example, if we assume that SPAM noise (i.e. that which we would attribute to the initial and final single-qubit-only layers) is global depolarizing (as done for MRB i ]) with polarization p, then the multiplicative factor in Eq. (13) is A p4. ≤ system into smaller subsystems and reconstruct the global n-qubit purity from the local purity of pairs of contiguous subsystems, like stitching together the bigger purity, with a number of measurements that overall grow polynomially in n; as a direct consequence, our protocol could be used to estimate global average noise coherence with a similar sample complexity.

[0169] Nevertheless, several things remain unclear, e.g., what classes of (Markovian) noise would such a method hold for, how could it be phrased beyond 1-dimensional qubit arrays, or how to certify the global estimations, among others. As we have argued, however, the average coherence of noise is a crucial figure of merit for benchmarking the error mechanisms limiting the performance of quantum computers, perhaps only standing after average layer fidelity in terms of importance as a benchmark, so it is highly important that the aforementioned hurdles are soon overcome.

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[0258] A Preliminaries

[0259] Here we detail and expand on some of the notation and technical details that, go into showing the results claimed in the main text. Much of the structure for the circuits entering the protocol is inspired by [ - ], so for this reason we stick to most of the original terminology and notation.

[0260] A.l Q-dist ribo ted circuits

[0261] We will consider a quantum system made up of n > 2 qubits and a pair of single-qubit and two-qubit gate sets, Gi and G2, respectively, with corresponding probability distributions 01 and Q2- We will then generate quantum circuits by sampling from gj ~ Qi and ga ^2, to generate parallel instructions on the n qubits as specified by a set IL(G) = {L{ = L^Lj2^}, where each e IL(Gi) and e 1L(©2) are instructions made up of only the single and two-qubit gates gi and g2 , respectively. To stress when some quantities are random variables, we use sans font when relevant, as in Lj being a random layer sampled from Q, which is such that

[0262] Circuits generated this way, here for the particular case G = (Gi, G2) and O = (Qi, Qa)) are said to be Q-distributed circuits, e.g., the circuit as stated in the main text as

[0263] Cni= LrnLm™i - • • Li, (18) where each Lj G IL(O) sampled according to 0, and the right-hand-side is read from right to left (i.e., Cmis an operator acting onto quantum states on the left), is a Q-distributed circuit of depth m.

[0264] We point out that, in general, fi-distributed circuits can be constructed with an arbi- trary number of gate sets and their corresponding distributions. Moreover, while so far G1. G2 and QI, Q2 are completely up to the user to specify, in practice the main require- ments we will have on this choice are that, i) the resulting Q-distributed circuits constitute an e-approximate unitary 2-design, and ii) that in particular (Gj , Qi) is an exact unitary 2-design, e.g., the single-qubit Clifford group. As such conditions i) and ii) are considered a posteriori for a particular case of interest, we will have a general derivation and then these will be stated formally in point 3 of $ ami in § , respectively.

[0265] A.2 Ideal vs real implementations

[0266] While we do not stress this difference in the main text, here we will denote by UL an ideal (noiseless) unitary operator associated to a representation of the layer L, and by UL (P):= ULPUI, the action of its corresponding map or superoperator UL.

[0267] We write a composition of two maps A and B simply as BA to mean “apply map B after applying map M”, equivalent to the usual notations B o A, or B(A(x)).

[0268] Noisy implementations of layers will generally be denoted by a map < / >, e.g., the noisy application of a layer L is denoted by the Completely Positive (CP) map (which we also refer to as a quantum channel, or simply, a channel) d(£), which can be defined (see e.g.. §IV of [1]) such that <p{L) := £L UL, with EL being generally a CP map associated to L. A.3 The output states of (^-distributed circuits

[0269] We will consider a set of fl-distributed circuits of depth m, acting on a fiducial initial state (0), which in turn we randomize with a layer V E IL(Gi). That is, we will have circuits of the form CmV|0), which will always give a pure state output in the noiseless case. In the noisy case, however, we will have states of the form averaged independently over each of the ((-distributed layers Lj, and uniformly over the initial single-qubit gate layer V e E(Gi) with flfi. This noisy output can of course now be mixed, 2~At ≤r(p)y) 1. ≤

[0270] Our protocol focuses on efficiently estimating the purities of the states om, to in turn estimate the average unitarity of noise in the circuits Cm. In the following, we first study the behavior of the average purity of pmin increasing circuit depth m, and then we see that we can efficiently estimate this quantity employing Randomized Measurements (RM)s, allowing us to readily extract the average unitarity of noise in certain circumstances which we analyze.

[0271] B Purity of average. outputs of Q-distributed circuits

[0272] B.l Purity of the initial randomization

[0273] Let us consider first the case of no gate layers but just the noisy initial single-qubit Clifford layer, m = 0, then we have where we defined the noisy implementation of V as ^>(V) := £Spam Mz for some noise channel £spam explicitly standing for State Preparation and Measurement (SPAM) noise, and where we defined the initial randomized pure state as Zfo(|OXO|) := IX’XPl-

[0274] Noticing that the noisy outputs ESpam(|V4(X’l)arequantum states and thus Hermitian, and that we can define the self-adjoint map X^ of a quantum channel X by the property tr[AA(B)] := tr[dG(A)i?L we can write

[0275] Equivalently, the self-adjoint map of a quantum channel can be defined in terms of its Kraus operators as fVt(-) := X where {Kfl} the Kraus operators of X, so Eq. (21) in a sense is already measuring how different Tspamis from a unitary.

[0276] In general, the purity of a quantum state, p := 2V( IM’X'W’I)canbe written as tr(p2) := (y| ly), which corresponds to the gate fidelity of X'X with respect to the identity on the state |y). Thus, we can write Eq, ( ) as a gate-fidelity of Pspan-fospam with respect to the identity, averaged over all possible initial states |y). where here

[0277] F^(A’) := (^df(|X#X)l< (23) is the gate-fidelity of the map X with respect to the identity map on the state |y>).

[0278] B.2 Relation to average unitarity and average trace-decrease

[0279] Ultimately, however, the average loss of purity in the average state OQ must be related to some amount of non-unitarity of the noise, i.e., it is noise that decreases purity. This can be quantified by the average unitarity, ug, defined for a CP map E as which is so defined to account for the case of £ being trace-decreasing. When £ is Trace Preserving (TP) we have the relation £^ '£'(•) = £^£(- — l / 2n), otherwise, however, for any quantum state p, where

[0280] S4£) := tr[£(|UXrt|)l?(27) is a trace-preservation measure of the map £ for a given sample IX5).

[0281] We can then write the average purity of po as which we recall is defined as averaged over the states IX’XX’I = Mv(|OXO|), which are dis- tributed uniformly and independently on the n-qubits according to a distribution Qj on single-qubit gates Gi.

[0282] When the noise channel £ is TP, the rightmost summand in Eq. ( ) becomes a term equal to — 2-n, and in that case the average unitarity is just a re-scaling of the average gate-fidelity of the form and thus also the average purity in Eq. ( ) in this case would become B.3 The case of global depolarizing noise

[0283] A particular case of a quantum channel for modeling noise that is both easy mathematically as in interpretation is that of global depolarizing channel, where 0 p ≤ 1 ≤ is the probability for the input state to remain the same, and which can be directly defined in terms of the channel by the so-called polarization 7,

[0284] 4"Fent(£) - l ,z

[0285] 7(«) := fog '(32)where here is the so-called entanglement fidelity (equivalent to the purity of the so-called Choi state of £). Because maximally entangled states can be written in the Pauli basis as jip)(Tj = 4~~nErePauiin ® PT, this is equivalent to where we identified the matrix £, with elements M = Matrix representation of 8.

[0286] For a depolarizing channel, in particular, h(M = p.

[0287] Also, a composition of depolarizing channels with the same polarization is just another depolarizing channel as

[0288] Given the above property, for a depolarizing channel, we also have for any state |fo). To stress this independence of the state tip we drop the subindex in the last line, which simply means u(Ep) := u^(Fp) for an arbitrary pure state |g>).

[0289] Notice that, £PUQ = UQ EP, or more generally,

[0290] £pX = X£p, (38) whenever X is a unital and TP quantum channel. Furthermore, for any unital and TP channel X, the unitarity uy(EpA) only depends on2X), which satisfies

[0291] F£(X%2X) = (TIMM MIMMM

[0292] - u(M (Fy(MA) - 2”"n) + 2M (39) and so i.e. the unitarity this way for depolarizing channels factorizes.

[0293] B.4 Average purity in the many-layer case

[0294] We now first consider adding a single layer of gates according to the distribution fl, and evaluate the purity of pi ,

[0295] This can then be generalized directly to the case of m layers as and thus by means of Eq. ( ) , this is equivalent to + trace decreasing terms, (43) where the trace decreasing terms are terms in S that can be read directly from Eq. (26) . One can express this way the most general form of the purity of the average state ornin circuit depth m.

[0296] However, we care, of course, for the situation in which the decay is exponential in m with the unitarity as the rate of decay.

[0297] 1. We may consider first that the noise is approximately gate and time independent, in which case at most we can get a double exponential in u(£) and S(£), as in [ ]. This approximation can be formalized by assuming that the average over time steps and gates is the leading contribution to the actual noise; see e.g., [ ].

[0298] 2. If we consider noise that is only gate-independent but that is unital and TP, we have and we can average recursively over each layer. 3. This circuit averaging, however, becomes significant if the layer set at least forms an e- approximate unitary 2-design1 1, to the effect that EL~Q U[£^£U\_ « Epfor a global depolarizing channel Ep2 with polarization where in the first line {K^} are the Kraus operators of £ [ ], or equivalently in the second line with E being a matrix representation of 8li!, or in terms of the entanglement fidelity through Eq. ( ) in the third line, or finally through the average gate- fidelity [ j, as we write in the main text. Thus, if we further consider fl- distributed circuits that form an approximate unitary 2-design, we may write where the approximation here neglects the small e multiplicative corrections: due to the property in Eq. ( ), and where we intentionally defined the polarizations to be squares, given that p2= u(EPi) For approximately time-stationary noise, to the effect that all polarizations arc equal, p^ = ■ ■ ■ - pm— p, this is equivalent to the exponential decay («) where A = 2~n(2n is a SPAM constant. Of course, given that u(Ep) = p2, this unitarity can also be expressed by u(£), as done in Eq. ( ) in the main text.

[0299] C Operational estimation of the purity

[0300] Now we know how the purity of the noisy average state pmin Eq. ( ) behaves in terms of the average unitarity of noise with respect to circuit depth m. The task is then to estimate

[0301] 11A unitary t-design is a probability measure p on the d-dimensional unitary group U(d) (or a subset thereof), such that T^'1= T^, where dp,(rf)?7®t(-)(E®i)t is a / --twirl over the unitary group. There are several nonequivalent, albeit often related, ways of approximating unitary designs, here we employ that in [ , ], where (1 — £)T^t2>A b (1 + 8)7^ with A 4 B here if and only if A — B is

[0302] CP, which can be interpreted in a straightforward way as having a small multiplicative factor of difference when measuring a state acted on with vs with Finally, for the case t ~ 2, the 2- twirl on a CP map <f> can be stated in terms of its matrix representation 4? through 7ff2^(4?), equivalent to the expression dp(U)U^(U plA)U for any state p.

[0303] 12 / S(E) X di \ It is possible to identify X = I J, '5j , where S'(£), ftfidi, Xnand AMitai are trace-decreasing,

[0304] \ zV n X unital / state-dependent leakage, non-unital and unital components (of dimensions 1, 1 x (4n— 1), (4n— 1) x 1 and square (4n— 1)), respectively. This leads to the expression in

[0012] for the average unitarity.

[0305] 13We notice that this does not mean that Spcorresponds to (approximately) the average of a single £ (which only occurs if 8 itself is already depolarizing) . this quantity via RMs.

[0306] C.l Purity estimation through randomized measurements

[0307] RMs found some of their first applications in estimating the purity of the quantum state of a subsystem | , ] Specifically, here we will focus on the results of [ , ], whereby it is shown that for a reduced quantum state p& = tr B (PAB)- from some composite closed quantum system AB, where here we assume A to be a n-qubit quantum system, its purity can be expressed as where s and s' are n-bit strings, 7i(s, s') is the Hamming distance (number of distinct bits) between them, Eu~Haar denotes uniform averaging with the Haar measure over the local unitaries id the

[0308] Pu(s) (S| WU(PA)|S), (49) are noiseless probabilities of observing the n-bit string s upon applying the unitary U on the reduced state p / \.

[0309] The main advantage here is that the quantum component of estimating purity this way is just the probabilities in Eq. ( ), which simply means applying a product of random unitaries and measuring each on a user-specified basis. The rest, i.e., fully estimating Eq. ( ) is purely classical post-processing of said probabilities. For n qubits, the number of distinct terms in the sum over n-bit strings in Eq. ( ) is 2n-1(2n— 1), since the Hamming distance is symmetric, h(s, s') = h(sz, s); however, this sum is only a function of the probabilities Pg, so it could, for example, be computed in parallel.

[0310] C.2 Purity of many layer Q-distributed circuits through randomized measurements

[0311] Let us again begin by considering the case m = 1, of a single random layer, so that the equivalent of the probability in Eq. ( ), by measuring the state pi with a RM using a random unitary W E E(Gi) over (li, reads as where we again defined Mv(|0)(0|) = |'X)(X’I38 arandom (according to a uniform U\j ~ fR) pure state, and identified each Esparnas the noise contribution to SPAM; in the penultimate line (purely for notational reasons) we defined ^(W) := and in the last line,

[0312] 14That is, Enaar here refers to a uniform and independent average over all the local unitaries Ui with the Haar measure, i.e., Enaar ~ ErnEr / g ■ ■ - Erjw Putting this together with the probability of getting another n-bit string s' with the same layer sequence, where in the last line we defined Pq(-) := iV’XX’l ’ IX’XVX •

[0313] Now we can exploit the identity derived in [ ], where here Fentis entanglement fidelity, defined in Eq. ( ), and U\j is the noiseless unitary map corresponding to the product U = Uj with local, single-qubit random unitaries Uj. Since this expression requires two copies of Zfo, where only the individual U, should be Haar random, it suffices for these (as opposed to the global unitary) to belong to a unitary 2-design, e.g., the single-qubit Clifford group.

[0314] We emphasize then, that we will require the RMs to be randomized by a layer made up of uniformly random single-qubit Clifford gates; while in the main text we fix this to correspond to Gi and Qi, all other layers in principle can be chosen arbitrarily (with the reasoning of having some better choices than others detailed in the previous sections).

[0315] We can now apply Eq. ( ) on Eq. ( ), so that where the second line follows directly from the definition of entanglement fidelity. This generalizes to any circuit depth m, to which coincides with Eq. ( ) up to £spamw•

[0316] Now similar to point 3 in § above, we mainly care about an exponential decay, which occurs for Q-distributed circuits being an approximate unitary 2-design and noise being approximately gated-time independent, unital and TP. Now we will have for A 2~n(2n up(£spamv), where Egspamwis the contribution from the layer average of £Spamw; of course both unitarity terms in A cannot be distinguished and can simply be interpreted as an average unitarity of SPAM.

[0317] The result in Eq. ( ) corresponds to that with an exact average over layers and initial states; this average can be approximated numerically with Eq. ( ) given a number Aw of measurements, Crnof Q-distributed circuits, and Afo of initial randomization samples, respectively. D Operational estimation of fidelity through randomized measurements

[0318] Since estimating the average unitarity is really useful only when informed by the average fidelity, we now consider estimating the average layer fidelity of the same O-distributed circuits, once given a set of randomized measurement probabilities P(j.

[0319] Naturally, if the noiseless output is a state | ip) , Eq. ( ) can be modified to account for its fidelity with respect to a- mixed state p as where p^deai^ jsthe probability of observing a n-bit string syea| with a randomized mea- surement defined by U on |<pX<ph that is

[0320] Notice that this assumes that we can efficiently compute the n-bit strings, and thus the probabilities, corresponding to the noiseless outputs. This generally implies a limitation in the gate sets that can be employed to only Clifford gates, however, for mid-scale estimations the ideal probabilities can still be computed without a major overhead when the gate set contains non-Clifford gates to be sampled with a. relatively low probability.

[0321] In our case, we want to estimate the quantity where

[0322] Thus, consider first the case m = 1; we have which follows as in Eq. ( ), where in the last line we defined Ppip) := IX’XVl ' I’ / ’X'Cl- Similarly, we can follow all steps from Eq. ( j to Eq. ( ) in an analogous way to obtain which is simply an Randomized Benchmarking (RB)-like fidelity decay. Similarly here, if the O-distributed circuits approximate a unitary 2-design. to the effect that EL~Q ~ Ep.;, i.e., the average noise is approximately a global depolarizing channel with p.t- = (2nF(E) — l) / (2n— 1), we end up with which is a RB-like decay of the state fidelity in the circuit, depth m, where we defined p'm= PmPSpamwfor some ftpamw- Further, for time-independent noise, such that pi = pj = . . . = pm= p, this is an exponential decay in m, 2 — 1 which can similarly be writen in terms of entanglement fidelity of £, or in terms of its matrix representation, similar to the usual R.B constants for unital SPAM.

[0323] This means we can generate a set of random fl-distributed circuits, attach some set of randomized measurements, and use the probabilities to simultaneously estimate average layer fidelity and unitarity.

[0324] E Unitary 2-designs

[0325] Here we refer to both assumptions, , as written in the main text in § for an exponential decay of the average sequence purity in circuit depth.

[0326] While assumption is beyond the choices that the user executing the protocol can make, assumption Ag. is important to make the protocol 3 adhere to the exponential decay of main Result 1 in § , but it also sets a practical limit in scalability to mid-scale systems, which we discuss below.

[0327] There is, furthermore, a fundamental limit to scalability imposed by the sample complexity of estimating purity via RMs, i.e., the number of experimental samples, Ameasin Eq. (10), needed to estimate purity within a given error. This is discussed in the main text in § , and it compounds with the sample complexity of the fl-distributed circuits, related to assumption

[0328] The need for unitary 2-designs in RB-based techniques stems from the fact that, by definition, these inherit the property of fully random (i.e. Haar distributed) unitaries whereby their second moment is described by a depolarizing channel, which requires a single parameter to be fully specified. This implies that figures of merit, such as average gate fidelity or average gate unitarity can be encapsulated into such polarization parameter (once assumption is satisfied) given circuits that generate a unitary 2-design.

[0329] Loosely speaking, a unitary 2-design is a probability distribution p on the unitary group, or a subset thereof, satisfying for any quantum channel X.

[0330] The action Ey~g VT(-)V is referred to as a 2-twirl with p. That is, the 2-twirl with p reproduces the second moment of the whole unitary group with the Haar measure. That an ensemble of unitaries constitutes a 2-design is particularly useful because it is known that doing a 2-twirl with the Haar measure on a CP map, reduces it to a depolarizing channel [ ], i.e., also for a 2-design V'iR’y(-) = p(-) + (1 — p)l / 2”, where here p = (2raF(A) — l) / (2n— 1), where F(ff) is the average gate fidelity of X. Both Eq I ) and Eq. ( ) are obtained using this relation.

[0331] The notion of a unitary 2-design can further be relaxed by having equality up to a small e according to a given metric, to the effect that the 2-twirl is also close to a depolarizing channel. More precisely, denoting a 2-twirl by A;i(-) we say that V is an e-approximate unitary 2-design, if

[0332] (1 — e)Anaar Ag =3 (1 + €)Anaarj (64) where here is semidefinite ordering in the sense that X =3 T if and only if X — is a CP map. This definition was put forth (for the more general setting of t-designs) in

[0063] and has the particularly simple interpretation of having a small multiplicative factor 1 ± e of difference when measuring a state acted on with a channel twirled as opposed to with Ahaar Thus, the approximation sign in both Eq. (13) and Eq. (15) refers to these small multiplicative factors in the 2-design approximation, exponential in m.

[0333] A well-known example of an exact 2-design is the n-qubit Clifford group [ ] : it is also simultaneously a strong reason why standard "JClifford RB works so well, and the culprit (in practice) of why it does not scale in n. Techniques akin to standard Clifford RB, such as Direct Randomized Benchmarking (DRB), Mirror Randomized Benchmarking (MRB) and Binary Randomized Benchmarking (BiRB) resolve such scalability issues by, among other things, retaining some of the randomness through Q-distributed circuits. Rather than requiring an approximate unitary design property, e.g., MRB [ ] and BiRB [ ] require a scrambling property in the sense that Pauli errors get quickly spread among other qubits before other Pauli errors occur. This can. be stated in terms of an entanglement fidelity for non-identity Paulis P, P\ with action P(-) = P(-)F and P / (-) = P'QP', such that for an expected infidelity of the layers a, there is k 1 / a and a § 1 with where here CQ — U{L^ - ■ - Lp), with U the unitary map of the sequence of noiseless layers Lfc • ■ • Li . Similarly, DRB uses generators of unitary 2-designs as the benchmarking gate set and then requires that these constitute a sequence- asymptotic unitary 2-design [ ].

[0334] While the scrambling condition in Eq. ( ) is stated for Pauli noise channels, the choice of (Gi, flj) being the uniformly distributed single-qubit Clifford group has the effect of projecting any CP map (modeling Markovian noise) to a Pauli channel (i.e., a tensor product of depolarizing channels) since each Clifford group constitutes a unitary 2-design on the respective qubit.

[0335] The reason approximating a unitary 2-design is a stronger condition than scrambling, as defined by Eq. ( ), is because it would require <5 shrinking as (9(4~n)1' . Nevertheless, while we directly assume that our fl-distributed circuits approximate a unitary 2-design, for a mid-scale of tenths of qubits the scrambling condition suffices, and indeed it holds in a similar way than it does in MRB or BiRB , as exemplified in § 5. Increasing the qubit count to estimate average unitarity would require establishing a scrambling condition, e.g., as for BiRB by employing stabilizer states and measurements, or otherwise relaxing the approximate design condition. loThis definition is further connected in [ ] to the usual diamond norm definition as implying that if V is an e-approximate unitary 2-design, also ||A(1— AHaaJ|02e. where jfVj* := supp||(2k ≤ ® X )p||i , where !|X||i = tr wXM denotes trace norm arid the supremum is taken over all dimensions d > 1 of the identity and corresponding density matrices p.

[0336] 16As can be seen in [ ] and [ ], there is a plethora of techniques under the term RB; by standard we mean single or two-qubit RB with corresponding Clifford gate sets, estimating the group’s average gate fidelity by fitting survival probabilities to an exponential decay in sequence length.

[0337] 1' This can be seen e.g., by assuming the W(Lfc • • ■ Li) generates a unitary 2-design and using Eq. ( )

[0338] 18A difference of our technique with MRB is that we do not employ a mirror structure, while one with BiRB is that we do not employ initial and final stabilizers. F Numerical Addendum

[0339] F.l Comparison of layer fidelity estimation with MRB

[0340] Circuit Layer Depth (log)

[0341] (b)

[0342] Figure 7: Comparison with MRB: Fidelity decays for Q-distributed circuits considering a uniformly distributed single-qubit Clifford group and a CZ gate on a line of 9 qubits with nearest-neighbour connectivity. In both figures, individual plots correspond to different number of qubits, small points represent individual state fidelity outputs at a given depth (for (a) taken over Median of Means (MoMs) estimators), bigger points represent average state fidelities at corresponding depth, and lines correspond to least-square fits of the corresponding averages to the exponential decay model ( ). In (a) we show the outputs with our technique using sample parameters Arcy = 12, JVW= 9, Wmeas= 210and K = 9 MoMs, while in (b) we show the outputs of MRB using 20 circuit samples with 20 set of Pauli layer samples each.

[0343] The ^-distributed circuits we consider are a primitive component in MRB [ ], thus the average fidelity estimations from both our technique and MRB should agree. There are two main differences between both procedures: one is that we do not construct a “mirror” circuit, i.e., we do not. append the circuit made up of the inverses of the original ^-distributed circuit, and the second is that we do not interleave layers of random Pauli gates. Both have their reason for being there or not in each technique, while we are able to estimate average unitarity too. this comes at a sampling complexity cost that requires exponentially many more samples to increase accuracy.

[0344] With a setup as in the main text, i with the only difference of considering a line topology, with qubits arranged in increasing order from 1 to 9 and nearest-neighbour connectivity, one can obtain the results of Fig. 7. Both techniques agree, with the biggest discrepancy occurring for a larger number of qubits, and with an increasing uncertainty in the case of our RM technique.

[0345] G UnitanW and Randomized Compiling

[0346] To begin with, let us label the n-qubit Pauli group Pn, i.e., the group made up of all n-fold products of Pauli operators (1, X, Y, Z), a t a be a 2n-bit string, a = 0402 . . . «2n, and define where Xj and Zj are single-qubit Pauli operators acting on the jthqubit and T := ®ji=i (0 0)such that is ensured to be Hermitian. The Pauli group Pnis then made up of all such Patogether with their products and their overall phases e.g.,

[0347] IP] = {±1, ±111, ±X, . . .. ±iZ}. In particular, denoting any other 2n-bit strings with bold lowercase letters, we will make use of the property

[0348] WA - ^l)'a^PbPa, (67) where

[0349] (a, b } aT(Y + TT)t> mod 2 which in particular is such that too)

[0350] Let us then write a generic n-qubit noise channel (i.e., some CP map) in the so-called y-representation. F( XabPa(;)Pb-. where y is a positive Hermitian matrix. We can also relate this representation to the so-called Pauli Transfer Matrix (PTM) representation, which is a 4n-square matrix with real entries given by which is not quite informative by itself; however, a particular case where this representation is useful is for the case of a Pauli channel, which has for y a diagonal matrix, implying that its PTM is diagonal too. A Pauli channel can be enforced on any channel by twirling it with the Pauli group, i.e., which gives where we made use of both Eq. ( ) and Eq. ( ), so that indeed the PTM of the twirled map EQ, is manifestly diagonal with eigenvalues Aa:= where we write the diagonal elements with a single index as aa:= Xaa-

[0351] The diagonal elements a are the so-called Pauli error probability rates, conversely related to the PTM eigenvalues as aa= and in terms of these, the trace non-increasing condition translates to aa 1, satu ≤rating for E being TP.

[0352] If follows that the average gate fidelity of a Pauli channel is given by ) where in the penultimate line we highlighted that it is the same as for the raw (not-twirled) channel E, and where used Eq. ( ), with 0 being the 2n-bit string with all bits being zero, and in the last line the case=1 f°r£ being TP; such expression makes it manifest that average gate fidelity captures only the information of the probability of any Pauli error happeningJ't.

[0353] 19This does not mean that, average gate fidelity is insensible to coherent errors, but rather that it is only partially so through their contribution to the diagonal elements in the PTM. Conversely, the unitarity of the Pauli channel is given by which is minimal in the sense that for a non-identity it corresponds to a purely stochastic noise channel, as it only involves the diagonal y-matrix terms, the Pauli error rates, involved in the average gate-fidelity. Clearly, when the distribution of errors is uniform, i.e.. all ay = 2"nso that the channel is maximally depolarizing, the unitarity vanishes. As opposed to average gate fidelity, the average unitarity of the raw channel £ would capture all elements of its PTM. We can further lower-bound this as follows, so together with inequality ( ) in the main text, we can express the unitarity for the Pauli-twirled channel £Qas where here / (£) = (2”F(£) — l) / (2n— 1) is the noise-strength parameter in Eq. in the main text, and f(8) := 1 — F(£) is the average gate infidelity of £.

[0354] The lower bound in Ineq. saturates for depolarizing noise, i.e., when all non- identity Pauli error rates are the same, while the upper bound overestimates the unitarity for Pauli noise through cross-products of noil-identity Pauli error rates. Thus, the upper bound serves as a proxy for whether noise has coherent components, if it is above it, or whether it is potentially Pauli, if it is under it.

[0355] Of course, in practice, we cannot directly twirl noise quantum channels but we need a way of achieving it by manipulating the noisy gates in question. Whenever the ideal (or target) gates are Clifford, a Pauli twirl can be enforced by a so-called ^-twisted twirl i i. where the outer Pauli operators in Eq. (71) are acted on (or twisted) with the ideal Cliffords. This concept of a twisted-twirl, together with employing random samples instead of all the 4" possible Pauli operators, are at the core of so-called Randomized Compiling (RC), where we would then effectively have

[0356] N samples 4 ( ' ) w E P£(P W)P, (77) A P~T. where here P ~ Pauli means P are Pauli samples drawn uniformly at random with no repetition from Pn. and the sum runs over N 4rasuch s ≤amples. In practice, this random- ized sum would be computed by measuring and then averaging many circuits where the noisy gate is twisted-twirled: considering that the noisy gate may be transpiled into single and two-qubit gates, the way this is usually done is by precisely compiling the single-qubit gates with the Paulis in each random sample, with the overall gate being logically exactly the same. Since also in practice the Pauli operators will be noisy, care has to be taken not to increase the overall number of gates that are executed. In general if the Paulis them- selves have coherent noise, perfect diagonalization would be achieved strictly for N — > oo by sampling with repetition.

[0357] The corresponding PTM of £$(■) has components where now we can only approximate the property in Eq. ( ) with the N random samples of bit strings q: in the infinite sample limit, N — > cc, ,or if exactly each distinct bit string is sampled once, the term within parenthesis goes to Scd. Since (q, 2c) = 0 for any pair of bit strings q and c, the sum over q when c = d equals 1 for any number of samples; this just reflects the fact that twirling (or R.C) leaves the diagonal of the PTM invariant. Thus, we can write where here as before, is the perfectly Pauli-twirled channel in Eq. (71), and where we defined which correspond to the off-diagonal elements of the PTM upon the twirl action of an element Pq. The effect of N = 1, a single randomization, will simply be to either change a sign of an off-diagonal element or leave it as it is. For an even number N of samples, the off-diagonals will either vanish or be suppressed in magnitude as integer multiples of 2 / IV (with ratio less than one). On the other hand, for odd N, all off-diagonals vanish in magnitude as integer multiples of 1 / N,

[0358] Finally, then where similarly now, the off-diagonal contributions will vanish as (small) multiples of 1 / N2.

Claims

PATENT CLAIMS1. Method for estimating an average contribution of coherent noise associated to a set of operations of a quantum processing device, said quantum processing device including a plurality of n 2 ≥ qubits, means for applying a set G of quantum gates and means for applying measurements to the n qubits, said method comprising: for each of a plurality of different circuit depths m 1 , genera ≥ting a plurality of quantum circuits Cm, each comprising m sets of randomly sampled instructions with gates of said set G of quantum gates, for each quantum circuit Cmof circuit depth m applying said quantum circuit Cmto a random initial state pvof the n qubits to thereby obtain an output state prthereof, and ^m,V applying said measurement means to the n qubits so as to provide an outcome of a randomized measurement of said output state; calculating, by classical postprocessing of the measurement outcomes, an estimator valueof an average purity of all output states pCmVof the circuits of depth m on the basis of the measurement outcomes of the randomized measurements; conceiving a fitting model for a data set {(m, $m)} of the estimator values of the purity and the circuit depth m, wherein a unitarity u of the noisy gates of the randomly generated circuits Cmis a parameter of said model, said unitary being indicative of the average contribution of the coherent noise; fitting the model to the data set to thereby obtain an estimation of the value of the unitarity u.

2. Method according to claim 1 , wherein said measurement means is operative to measure in the computational basis |s) = |s1(...,sn), st= 0,1, of the n qubits, said method further comprises generating a plurality of Nw randomization circuits W, each comprising a randomly sampled randomization instruction with gates of the gate set G, and wherein providing the randomized measurement outputs for the output state prcomprises for^m,V each randomization circuit W applying said randomization circuit to said output state p ^rm,V followed by applying said measurement means to the n qubits to thereby measure in the computational basis to obtain a measurement outcome s = (st, ...,sn).

3. Method according to claim 2, wherein for each circuit depth m the average purityof all output states pLrm,V of the circuits Cmof depth m is estimated on the basis of the estimator values of the outcome probabilitiestherandomized measurements according towherein h(s,s') is the Hamming distance between the measurement outcomes $ = (sv... , sn~) and s' = (s^, ... , sn'~), Nvis the number of distinct random initial states pvfor the circuit Cm, Nwis the number of distinct randomization circuits, and NCmis the number of circuits Cmwith depth m.

4. Method according to anyone of the preceding claims, wherein said method further comprises:Estimating, by a simulation on a classical computer, for each quantum circuit Cma value of outcome probabilities of a noiseless randomized measurement applied to a noiseless output state obtained by a noiseless application of the quantum circuit Cmto a noiseless random initial state pv\ calculating, by the classical computer, for each circuit depth m, an estimator value of an average fidelityof circuits of depth m on the basis of the measurement outcomes of the quantum processor and the classically calculated noiseless outcome probabilities; conceiving a fitting model for a data set {(m, §m)} of the estimator values of the average fidelityof circuits of depth m and the circuit depth m, wherein an average gate fidelity is a parameter of said model; fitting the model to the data set to thereby obtain an estimation of the value of the average gate fidelity.

5. Method according to claim 4, wherein calculating the estimator value of the average fidelity of circuits of depth m comprises determining, by classical simulation, the probabilitiesof the noiseless measurement outcomes and calculating the average fidelity wherein h(s,s') is theHamming distance between the measurement outcomes s = (s1;...,sn) and s' = (s / , ... , sn'), Nvis the number of distinct random initial states pvfor the circuit Cm, Nwis the number of distinct randomization circuits, and NCmis the number of circuits G„with depth m.

6. Method according to anyone of the preceding claims, wherein the fitting model for the average purityis of the form Aum+ -^ = $m, wherein 0 A 1 is a ≤ cons ≤tant.

7. Method according to anyone of claims 4 - 6, wherein the fitting model for the average fidelity %mof circuits of depth m is of the form = afm+ — , where 0 a < 1 , an ≤d apolarization f =2n-iis defined in terms of the average gate fidelity F of an average layer noise with respect to the identity averaged over the Haar measure, F :=wherein £ is the average layer noise.

8. Method according to anyone of the preceding claims, wherein generating the quantum circuits Cmis such that they form an £-approximate unitary 2-design.

9. Method according to anyone of the preceding claims, wherein the set G comprises a first set Gi of single-qubit gates and a second set G2 of two-qubit gates.

10. Method according to claim 9, wherein each of the m sets of instructions comprises a first instructionwith gates of the first set G1 performed in parallel, and a second instruction L® with gates of the second gate set G2 performed in parallel, wherein the first instructions 1$ are sampled according to a first probability distribution Qi, and the second instructions L® are sampled according to a second probability distribution CI2, and the first and second instructions are subsequent instructions.

11. Method according to anyone of claims 9 or 10, wherein the set of single-qubit gates G1 is a unitary-2 design.

12. Method according to claim 11 , wherein the set of single-qubit gates G1 comprises the set of single-qubit Clifford gates.

13. Method according to anyone of claims 10 to 12, wherein the first probability distribution Qi is a uniform distribution.

14. Method according to anyone of the preceding claims, wherein said method further comprises generating for each quantum circuit Cma state preparation circuit VCmcomprising a randomly sampled state preparation instruction with gates of the gate set G, and preparing the random initial state pvfor said circuit Cmby applying said state preparation circuit VCmto a predetermined quantum state which is the same for all quantum circuits.

15. Method according to claim 14, wherein each state preparation instruction V comprises instructions with gates of the first set G1 performed in parallel and is sampled according to the first probability distribution Qi.

16. Method according to anyone of claims 2-15, wherein each randomization instruction W comprises instructions with gates of the first set Gi performed in parallel and is sampled according to the first probability distribution Qi.

17. Computing system, said computing system comprising a classical computer and a quantum processing device, said quantum processing device including a plurality of n 2 qubits, means ≥ for applying a set G of quantum gates and means for applying measurements to the n qubits; wherein said classical computer is operative to generate, for each of a plurality of different circuit depths m 1 , a ≥ plurality of quantum circuits Cm, each comprising m sets of randomly sampled instructions with gates of said set G of quantum gates; wherein said quantum processing device is operative to apply, for each quantum circuit Cmof circuit depth m, said quantum circuit Cmto a random initial state pvof the n qubits to thereby obtain an output state prthereof, and to apply said measurement means to the n qubits so as to provide an outcome of a randomized measurement of said output state; wherein said classical computer is further operative to calculate, by classical postprocessing of the measurement outcomes, an estimator valueof an average purity of all output states pCm Vof the circuits of depth m on the basis of the measurement outcomes of the randomized measurements and to fit a fitting model for a data set {(m, $m)} of the estimator values of the purityand the circuit depth m to the data set, wherein a unitarity u of the noisy gates of the randomly generated circuits Cmis a parameter of said model, said unitary being indicative of the average contribution of the coherent noise, to thereby obtain an estimation of the value of the unitarity u.

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