From randomized benchmarking experiments to gateset circuit fidelity: how to interpret randomized benchmarking decay parameters

Randomized benchmarking (RB) protocols have become an essential tool for providing a meaningful partial characterization of experimental quantum operations. While the RB decay rate is known to enable estimates of the average fidelity of those operations under gate-independent Markovian noise, under gate-dependent noise this rate is more difficult to interpret rigorously. In this paper, we prove that single-qubit RB decay parameter $p$ coincides with the decay parameter of the gate-set circuit fidelity, a novel figure of merit which characterizes the expected average fidelity over arbitrary circuits of operations from the gate-set. We also prove that, in the limit of high-fidelity single-qubit experiments, the possible alarming disconnect between the average gate fidelity and RB experimental results is simply explained by a basis mismatch between the gates and the state-preparation and measurement procedures, that is, to a unitary degree of freedom in labeling the Pauli matrices. Based on numerical evidence and physically motivated arguments, we conjecture that these results also hold for higher dimensions.

Recently it has been shown that RB experiments on an arbitrarily large number of qubits will always produce an exponential decay under arbitrary Markovian error models (that is, where errors are represented as completelypositive maps). This ensures a well-defined theoretical characterization of these experiments and enables an important test for the presence of non-Markovian errors, in spite of the gauge freedom between the experimental quantities and a theoretical figure of merit such as the average gate fidelity [18,19]. However, this theoretical advance still lacks a clear interpretation tying the experimental decay to a fidelity-based characterization of physical gate-dependent errors.
In this Letter, we show that in the regime of high fidelity gates on single qubits, such an interpretation does exist. Further we conjecture, based on numerical evidence, that such an interpretation extends to arbitrary dimensions. Consequently, this work provides an important tool for identifying and eliminating errors through examining the results of RB experiments. 1 SPAM stands for "State preparation and measurement".
Consider an ideal gateset G = {G} and its noisy im-plementationG = {G}. We denote a circuit composed of m elements byG m:1 :=G m · · ·G 2G1 . (1) For leakage-free RB experiments with arbitrarily gatedependent (but still Markovian) errors, the average probability of an outcome µ after preparing a state ρ and applying a circuit of m + 1 operations that multiply to the identity is [19] E Gm+1:1 Tr µG m+1:1 (ρ) = Ap m + B + (m). (2) Here, A and B are independent of m (i.e., they only depend upon ρ, µ andG) and (m) is a perturbative term that decays exponentially in m. By design, RB gives some information about the error rate of motion-reversal (i.e., identity) circuits composed of gateset elements. In this paper, we show how this information relates to general circuits. Consider the traditional notion of average fidelity for a noisy circuitC to an ideal unitary circuit C, where the integral is taken uniformly over all pure states. Equation (3) corresponds to the definition of the usual notion of average gate fidelity, but is instead formulated in terms of "circuit", which is to be understood as a sequence of elementary gates. We introduce this nuance to define a novel figure of merit, the gateset circuit fidelity, which compares all possible sequences of m implemented operations from the gatesetG to their ideal analog in G, Definition 1 (Gateset circuit fidelity). The case m = 1 yields the average fidelity of the gateset G to G. In general, the overall action of ideal circuits G m:1 is reproduced byG m:1 with fidelity F(G, G, m). In this Letter, we prove that for all singe-qubit gate sets with fidelities close to 1, the gateset circuit fidelity can be robustly estimated via RB experiments, for all circuit lengths m. We conjecture this result to hold for higher dimensions, based on numerical evidences and physically motivated arguments.

II. THE DYNAMICS OF THE GATESET CIRCUIT FIDELITY
It follows from the RB literature [1,5] that for gateindependent noise models of the formG = EG or G = GE, where E is a fixed error, the gateset circuit fidelity behaves exactly as where p is estimated through standard RB by fitting to eq. (2) with (m) = 0 and d is the dimension of the system. For gate-dependent leakage-free noise models, eq. (5) generalizes to where the fidelity on the traceless hyperplane is similar to the gateset circuit fidelity, but is averaged over the traceless part of the pure states, ψ tr = ψ − I/d: Under gate-dependent noise, f tr (G, G, 1) could substantially differ from p [18]. For example, letG = UGU † for any fixed non-identity unitary channel U so that f tr (G, G, 1) < 1. Then the composition of noisy gates with themselves appears to be perfectly self-consistent and so all motion-reversal circuits would exactly implement the identity gate. Such operation is independent of the circuit length m and so eq. (2) is also independent of m; therefore p = 1 > f tr (G, G, 1). This apparent disconnect arises due to a basis mismatch between the bases in which the SPAM and the gateset are defined. Having a perfect gateset circuit fidelity is appropriate in this example because the only possible circuit errors arise from the mismatch between the gateset and SPAM procedures.
In appendix A, we show that the disconnect between p and f tr (G, UGU † , 1) strongly depends on the choice of basis U. That is, we prove where U is a physical unitary channel (see theorem 5). D(m, U) is typically negligible or becomes rapidly negligible as it is exponentially suppressed in m. This means that the relative variation in f tr as the circuit grows in length, depends weakly on the choice of basis U. More precisely, δ(m, U) is composed of two factors: the first one decays exponentially in m and is at most of order (1 − p) m/2 , while the second carries the dependence in U; the existence of a specific choice of U such that this last factor becomes at most of order (1 − p) 3/2 is proven in the single-qubit case (appendix B), and conjectured to hold in general. The explicit behaviour of δ(m, U) given a numerically simulated gate-dependent noise model is illustrated in fig. 1. Consequently, the gateset circuit fidelity can be updated with a good approximation through the recursion relation (10) To provide insight on the total value of the gateset circuit fidelity given a circuit's length m, we need a stronger relation between the RB estimate of p and the gateset circuit fidelity. Fortunately, the freedom in the choice of ideal gateset can be fixed in a way that allows us to estimate the total change in gateset circuit fidelity for arbitrary circuit's lengths. In appendix B, we prove that the disconnect between p and f tr (G, UGU † , 1) under general gate-dependent noise is almost completely accounted for by a basis mismatch.

Proposition 2.
For any single-qubit noisy gatesetG perturbed from G, there exists an ideal gateset UGU † , where U is a physical unitary, such that In fact, we conjecture this result to hold for any dimension, or at least for most realistic gate-dependent noise models. To grasp the physical reasoning behind this, we refer to the end of appendix B, as it rests on some prior technical analysis. The extension of proposition 2 to 2qubit systems is supported by numerical evidences (see appendices B and B and section II) To reformulate the result, the family of circuitsG m:1 built from a composition of m noisy operationsG ∈G mimics the family of ideal circuits UG m:1 U † with fidelity In the paradigm where the targeted operations G ∈ G are defined with respect to SPAM procedures, U captures the misalignment between the basis in which the operationsG ∈G are defined and the basis defined by SPAM procedures. This goes farther: consider an additional gateset, for which the targeted operations H ∈ H are also are defined respect to SPAM procedures. From proposition 2, there exists a physical unitary V for whichH m:1 imitates the action of VH m:1 V † with fidelity FIG. 1: Absolute value of the deviation δ(m, V) , described in eq. (9) (also see eq. (A16)), as function of circuit length m with noise model generated byG (16)). The red points are obtained with the choice of basis V = I, while the blue points are obtained with the choice V = U where U is found through eq. (15). The purple horizontal line corresponds to (1 − p) 2 , while the green line corresponds to (1 − F(G, G, 1)) 2 . For both ideal gatesets G and UGU † , the deviation becomes quickly negligible as the sequence length increases. In fact, in the case V = U (blue points), the deviation is always below . U † V captures the basis mismatch between the gatesetsG andH. Such a non-trivial mismatch could easily be imagined if, for instance, gates belonging toH were obtained through a different physical process thanG, or calibrated with regards to alternate points of reference.

III. CALCULATING BASIS MISMATCHES FOR SPECIFIC NOISE MODELS
We now discuss how the basis mismatch can be calculated for specific noise models, whether from numerical simulations, analytic approximations, or tomographic reconstructions. As shown in theorem 5 and eq. (8), the total change of gateset circuit fidelity depends on the physical basis in which the ideal gateset is expressed. In the single-qubit case, we showed the existence of a physical basis U that reconciles f tr (G, UGU † , m) with p m through proposition 2. One might suspect that the unitary U can be found through the maximization of the gateset fidelity: and indeed this would handle noise models of the form G = UEGU † , as However, this hypothesis fails for simple noise models of the formG = UEGU, where This case study is informative as these two last noise models share something in common: there exists a choice of unitary that cancels the noisy map on the right of the noisy gateset. Although such exact cancellation is not always possible, we now show that a close approximation is sufficient. Consider the slightly more general noise model of the formG = E L GE R , where we allow fixed but arbitrary error maps to the left and the right of an ideal gateset. It is easily shown that RB will yield p m = f tr (E R E L G, G, m), since E R E L is the effective error map between two otherwise perfect implementations of the gateset elements. In the single-qubit case (and for many, if not all physically motivated higher dimensional noise models) there exists a unitary operation U such that (see appendix B). That is, the fidelity of the map between two noisy gatesets can be seen as the gateset circuit fidelity between a noisy gateset and an ideal one. A choice of such physical unitary is which essentially cancels the unitary part of E R 2 . For more general gate-dependent noise models, the idea remains more or less the same. As shown in appendix B, the right error E R is replaced by its generalization, the 4 th order right error E U cancels the unitary part of the 4 th order right error.
This provides a means to guide the search of the appropriate ideal gateset of comparison UGU † given a numerical noise modelG. Indeed, the 4 th order right error is easily found, either by direct computation of the average E G 4:1 †G 4:1 , or more efficiently by solving the eigensystem defined in eq. (A7a). The optimization defined in eq. (15) can be solved via a gradient ascent parametrized over the d 2 − 1 degrees of freedom of SU (d).
In the single-qubit case, the optimization procedure can be replaced by an analytical search. Given the process matrix E With this at hand, we performed numerically simulated RB experiments under gate-dependent noise models. Each of the 24 Cliffords was constructed by a sequence of X and Y pulses, G x = P (σ x , π/2) and G y = P (σ y , π/2), where P (H, θ) := e iθH/2 .

IV. CONCLUSION
RB experiments estimate the survival probability decay parameter p of motion-reversal circuits constituted of operations from a noisy gatesetG of increasing length (see eq. (2)). While motion-reversal is intrinsic to the experimental RB procedure, the estimated decay constant p can be interpreted beyond this paradigm. In this Letter we have shown that, in a physically relevant limit, the very same parameter determines an interesting figure of merit, namely the gateset circuit fidelity (defined in eq. (4)): as a random operation fromG is introduced to a random circuit constructed from elements inG, p captures the expected relative change in the gateset circuit fideilty through eq. (10).
It is also possible to characterize the full evolution of gateset circuit fidelity as a function of the circuit length. In this Letter, we have also demonstrated that given a single-qubit noisy gatesetG perturbed from G, there exists a physical basis change UGU † such that the gateset circuit fidelity takes the simple form given in eq. (11). This gives a rigorous underpinning to previous work that has assumed that the experimental RB decay parameter robustly determines a relevant average gate fidelity (eq. (3)) for experimental control under generic gatedependent scenarios. We conjecture a similar result to hold for higher dimensions and provide numerical evidence and physically motivated arguments to support this conjecture. Given any specific numerical noise model G perturbed from G, we showed how to obtain a physical unitary U for which eq. (11) holds. The procedure can be seen as a fidelity maximation of the 4 th order right error acting on the gateset through a unitary correction (see proposition 3).
The introduction of such a physical basis adjustment is natural because it has no effect on how errors accumulate as a function of the sequence length. Rather, it only reflects a basis mismatch to the experimental SPAM procedures, and hence shows up as SPAM errors. This is in principle detectable by RB experiments but in practice not part of the goals of such diagnostic experiments. In particular, differences in the (independent) basis adjustments required for distinct gatesets will not appear in any characterization of the individual gatesets, but will be detected when comparing RB experiments for this distinct gatesets (e.g., comparing dihedral benchmarking and standard randomized benchmarking experiments which have distinct gatesets but share gates in common, or comparing independent single-qubit RB on two qubits -which has no two-qubit entangling gate -with standard two-qubit RB). We leave the problem of characterizing relative basis mismatch between independent gatesets as a subject for further work. In this section, we extend the standard RB analysis under gate-dependent noise provided in [19] in order to prove the claim from eq. (9) that standard RB returns the relative variation of the gateset circuit fidelity.
Let A be the Liouville matrix of a linear map A and Π tr (ρ) = ρ − I Tr ρ/d be the projector onto the traceless component. Let e j be the canonical unit vectors, A = j,k a j,k e j e T k , and Using the identity we have where the twirling superchannel [18,19,22] is and G tr = GΠ tr . Changing the gateset G to UGU † for some physical unitary U leaves Π tr = U Π tr U † . Therefore The spectrum of T is unchanged under the basis change G → U GU † . Moreover, its most important eigenvectors are as follows.
Since the noisy gatesetG is a small perturbation from G the spectrum of T will be slightly perturbed from {1, 0, 0, . . .}. Therefore (T /p) m approaches a rank 1 projector as m increases and so vec(B ∞ ) is a +1-eigenvector of T /p. The same argument applies to A T ∞ .
Lemma 4 allows us to write with ∆vec (B ∞ ) = vec † A T ∞ ∆ = 0. In eq. (A5), we can expand the vectors as and v(U), w(U) are implicitly defined unit vectors. Using this expansion together with eq. (A10) in eq. (A5) yields the following result: Theorem 5 (Total gateset circuit fidelity). The gateset circuit fidelity obeys where In [18,19] it is shown that standard RB provides an estimate of p. Notice that p is independent of the basis in which the ideal gateset of comparison, UGU † , is expressed.
From eq. (A14), it is straightforward to show that which is exponentially suppressed. We show in the next section that the eigenvalues of ∆ are at most of order √ 1 − p, which ensures a very fast decay, as shown in fig. 1. Equation (9) is in fact a reformulation of eq. (A16).

Appendix B: Varying the ideal gateset of comparison
In this section, we prove proposition 2 by determining how the basis U of the ideal gateset UGU † affects the coefficients in eq. (A14).
Let G be an ideal gate set defined with respect to the SPAM procedures. We can write the elements of a noisy gateset asG so that the perturbations δ G both capture the errors in the noisy gate and the mismatch with the targeted computational basis. Under gate-independent noise with no basis mismatch,G = EG and the infidelity of the perturbed operations I + δ A basis mismatch will change the infidelity of the perturbations roughly to r(UE) + r(U † ) for some unitary channel U, which will typically differ substantially from the fidelity inferred from the associated RB experiment.
Experimentally, such basis mismatches will be relatively small as operations will be somewhat consistent with SPAM procedures. Under this assumption, we now show that there exists an alternate perturbative expansion,G for which r(I + Eδ In appendix A, we showed that (T /p) n converges to a rank-1 projector. We now quantify the rate of convergence. By the Bauer-Fike theorem [23], for any eigen-value λ = p of T , This spectral profile implies that (T /p) n converges quickly to a rank-1 operator since the eigenvalues close to zero are exponentially suppressed. Hence, we can approximate the asymptotic eigenoperators defined in eqs. (A6a) and (A6b) as: In the simple noise model To pursue the analogy, we denote the m th order right and left errors as Combining eq. (B4) and eq. (B3), we get The structure of single-qubit error channels allows us to pursue a deeper analysis. It follows from the channel analysis provided in [25] that, for high-fidelity qubitchannels, the 3 × 3 submatrix acting on the traceless hyperplane can always be decomposed as where V is a physical unitary, and D is an incoherent process. Here we label a channel D incoherent if Expressing the 4 th order right error E R as allows us to maximally correct it through a physical unitary: Hence, since E R V † is a high-fidelity incoherent channel, the normalized trace of the composition Π tr E (4) is essentially multiplicative [26]:  This expression allows us to pick a better perturbative expansion than eq. (B1). Indeed, choosing ensures that the noisy operations I + δ have an gateset circuit infidelity which is more in line with the RB data: Iterating the analysis leads to This completes the demonstration of proposition 2.
Our current proof technique relies on the structure of single-qubit channels. For higher dimensions, we conjecture that an analog of proposition 2 holds, although the scaling with the dimension is unclear. Conjecture 6. If the fidelity of E (4) R is high, then ∃ a physical unitary V † s.t. E (4) R V † is incoherent.
As we now show constructively, conjecture 6 holds for physically motivated noise models composed of generalized dephasing, amplitude damping, and unitary processes. Under such noise models, for some unitaries U i and incoherent channels D i . The channel UDU † is incoherent for any physical unitary U, and the composition of incoherent channels is also incoherent, so eq. (B20) can be rewritten as E (B22)