Bounding the average gate fidelity of composite channels using the unitarity

There is currently a significant need for robust and efficient methods for characterizing quantum devices. While there has been significant progress in this direction, there remains a crucial need to precisely determine the strength and type of errors on individual gate operations, in order to assess and improve control as well as reliably bound the total error in a quantum circuit given some partial information about the errors on the components. In this work, we first provide an optimal bound on the total fidelity of a circuit in terms of component fidelities, which can be efficiently experimentally estimated via randomized benchmarking. We then derive a tighter bound that applies under additional information about the coherence of the error, namely, the unitarity, which can also be efficiently estimated via a related experimental protocol. This improved bound smoothly interpolates between the worst-case quadratic and best-case linear scalings for composite error channels. As an application we show how our analysis substantially improves the achievable precision on estimates of the error rates of individual gates under interleaved randomized benchmarking, enabling greater precision for current experimental methods to assess and tune-up control over quantum gate operations.


I. INTRODUCTION
The output of a quantum computer will only be reliable if the total noise in the whole computation is sufficiently small. This can be guaranteed if the noise on the individual components (i.e., preparations, gates and measurements) is sufficiently small compared to the length of the computation. The noise on the individual components can be diagnosed using gate set tomography [1,2]. However, gate set tomography is not scalable with the number of qubits. Furthermore, computing the exact effect of the noise on an algorithm is likely to be as difficult as classically simulating the algorithm.
Randomized benchmarking (RB) and variants thereof [3][4][5][6][7][8][9] can be used to efficiently, but partially, characterize the noise on individual gates. In particular, fitting the decay curve of an RB experiment provides an estimate of the average gate fidelity to the identity (hereafter simply the fidelity) of an error channel E, F (E) := dψ ψ|E(|ψ ψ|)|ψ (1) where dψ is the Fubini-Study metric. The fidelity is one if and only there is no noise, that is, if E is the identity channel. However, noise channels with the same fidelity can have a dramatically different effect on a circuit. For example, the infidelity r(E) = 1−F (E) grows linearly in the number of gates under purely stochastic noise (that is, noise that admits a Kraus operator decomposition using only Pauli operators) and quadratically under purely unitary noise in the limit of small infidelities [10]. However, realistic experimental noise is neither purely stochastic or purely unitary, but rather is some combination of the two.
In this paper, we study how the total infidelity of a circuit can be estimated from partial information about the components. We first obtain strictly optimal upperand lower-bounds on the total infidelity for all parameter regimes when only the infidelities of the components are known. These bounds are saturated by unitary channels and so grow quadratically with the number of gates. Moreover, because these bounds are saturated, they cannot be improved without further knowledge about the noise.
This worst-case growth of the infidelity is achieved by purely unitary channels. Intuitively, quantifying how far a channel is from purely unitary should enable an improved bound. One such quantity is the unitarity, which can be efficiently estimated using a variant of RB [7]. We prove that the unitarity can be used to smoothly interpolate between the quadratic growth of purely unitary noise and the linear scaling of purely stochastic noise by obtaining bounds on the total infidelity as a function of the infidelities and unitarities of the components. As a specific application, we show that the improved bound provides a substantial improvement to the accuracy of the estimates from interleaved RB [6].

II. NOISY QUANTUM PROCESSES
Noisy Markovian quantum processes can be described by completely-positive and trace-preserving (CPTP) linear maps E : D d → D d where D d is the set of density operators acting on C d , that is, the set of positive-semidefinite operators with unit trace. We denote quantum channels using single calligraphic capital Roman letters and the composition of channels by multiplication for brevity, so that AB(ρ) = A[B(ρ)]. We also denote the composition of m channels E 1 , . . . , E m by E 1: Abstract quantum channels can be represented in many ways, In this paper, we will use the Kraus operator, χ-matrix and the Liouville (or transfer matrix) representations. The Kraus operator and χ-matrix rep-resentations of a quantum channel E are respectively, where the A j are the Kraus operators, Note that we include the dimensional factor in the definition of the χ-matrix to be consistent with the standard construction in terms of unnormalized Pauli operators.
The Kraus operators can be expanded as A j = k∈Z d 2 B k , A j B k relative to B. Making use of the phase freedom in the Kraus operators (that is, A j → e iθj A j gives the same quantum channel), we can set B 0 , A j ≥ 0 for all j. We can then expand the Kraus operators as with v j 2 = 1, and α j can be chosen to be in [0, π 2 ] by incorporating any phase into v j . Substituting this expansion into the Kraus operator decomposition and equating coefficients with the χ-matrix representation gives and, in particular, Applying the trace-preserving constraint with B j , B k = δ j,k gives which then implies Alternatively, density matrices ρ and effects E (elements of positive-operator-valued measures) can be expanded with respect to B as ρ = j B j , ρ B j and E = j B j , E B j . The Liouville representations of ρ and E are the column vector |ρ and row vector E| = |E † of the corresponding expansion coefficients. The Born rule is then E, ρ = E|ρ . The Liouville representation of a channel E is the unique matrix E such that E|ρ = |E(ρ) , which can be written as TABLE I. Linear relations between the fidelity (F ), the infidelity (r), the RB decay rate (p), and χ00.
representation of any CPTP map can be expressed in block form as where E n ∈ C d 2 −1 is the non-unital vector and E u ∈ C d 2 −1×d 2 −1 is the unital block. The unitarity and RB decay rate can be written as with respect to the Liouville representation [7,11]. The RB decay rate p(E) and χ 00 are linear functions of the fidelity that can be more convenient to work with. The relations between the various linear functions of the fidelity used in this paper are tabulated in table I.

III. COMPOSITE INFIDELITIES IN TERMS OF COMPONENT INFIDELITIES
We now prove that unitary noise processes lead to the fastest growth in the total infidelity of a circuit. In particular, we obtain strict bounds on the infidelity of a composite noise process in terms of the infidelities of the components and show that the bounds are saturated by unitary processes for all even-dimensional systems.
We first obtain a bound on the infidelity of the composition of two channels that strictly improves on the corresponding bound of Ref. [11]. We also show that the improved bound is saturated for all values of the relevant variables. Therefore theorem 1 gives the optimal bounds on the infidelity of the composite in terms of only the infidelities of the components, and so obtaining a more precise estimate of the composite infidelity requires further information about the noise. We then obtain an upper bound on the infidelity of the composition of m channels that inherits the tightness of the bound for the composition of two channels.
We present the following bounds in terms of the χ matrix, though the results can be rewritten in terms of other linear functions of the infidelity using table I. For example, consider the composition of m noisy operations X i with equal infidelity, that is, r(X i ) = r. Then by corollary 2 and table I, the total infidelity of the composite process is at most which exhibits the expected quadratic scaling with m. Moreover, this upper bound is saturated and so cannot be improved without additional information about the noise.
Theorem 1. For any two quantum channels X and Y, Furthermore, for all even dimensions and all values of χ X 00 , χ Y 00 , there exists a pair of channels X and Y saturating both signs of the above inequality.
be Kraus operator decompositions of X and Y respectively. From eq. (4), we can expand the Kraus operators as where u j , v j ∈ C d 2 −1 are unit vectors and ξ j , θ j ∈ [0, π 2 ]. Then a Kraus operator decomposition of X Y is and so, by eq. (6), where β j,k = u j · v k and we have chosen the basis B to be Hermitian so that Tr B † j B k = Tr B j B k = δ j,k . By the triangle and reverse-triangle inequalities, for any α, β, γ ∈ C such that |β| ≤ 1, which then implies From eq. (6) and (8), so by eq. (17), using |β j,k | ≤ 1 and the nonnegativity of the trigonometric functions over [0, π 2 ]. Note that the above inequalities are saturated if and only if β j,k = ±1.
By the Cauchy-Schwarz inequality with the fact that all the quantities are nonnegative, where the second line follows from eq. (6) and eq. (8).
Applying this upper bound for X and the corresponding one for Y to eq. (19) gives the inequality in the theorem. To see that both signs of the inequality are saturated for all values of χ X 00 , χ Y 00 in even dimensions, let X = By eq. (6), χ which saturates the lower bound if the sign function is positive and the upper bound if it is negative.
the χ 00 element of the composite channel satisfies Furthermore, this bound is saturated for all even dimensions and all values of the χ Xi 00 satisfying eq. (22).
Proof. We can rewrite the lower bound in eq. (12) as Writing √ χ 00 = cos(arccos √ χ 00 ) and √ 1 − χ 00 = sin(arccos √ χ 00 ) and using standard trigonometric identities, the above becomes arccos χ X Y 00 ≤ arccos χ X 00 + arccos χ Y 00 , taking note to change the direction of the inequality when taking the arccos, which follows from eq. (22). By induction, we have for any set of m channels X i . Taking the cosine and squaring gives the bound in eq. (23). The saturation follows directly from the saturation of eq. (12).

IV. IMPROVED BOUNDS ON THE INFIDELITY USING THE UNITARITY
The bounds in theorem 1 and corollary 2 are tight for general channels if only the infidelity (or, equivalently, χ 00 ) is known. In particular, from eq. (11), the infidelity increases at most quadratically in m (to lowest order in r). However, the examples that saturate the bounds are all unitary channels. If, on the other hand, the noise is a depolarizing channel or a Pauli channel (that is, a channel with a diagonal χ matrix with respect to the Pauli basis), then the infidelity increases at most linearly in m to lowest order, that is The intermediate regime between Pauli noise and unitary noise can be quantified via the unitarity [7]. In particular, we define the (positive) coherence angle to be θ(E) = arccos p(E)/ u(E) .
As u(E) ≤ 1 with equality if and only if E is unitary, θ(E) ∈ [0, arccos p(E)] and That is, θ(E) quantifies the intermediate regime between Pauli and unitary noise for an isolated noise process. We now show that combining the coherence angle and the infidelity enables improved bounds on the growth of the infidelity. For example, for any m unital channels, or for any m single qubit operations X i , with equal infidelity r(X i ) = r and coherence angles θ(X i ) = θ, the total infidelity is at most plus higher-order terms in r and θ 2 by eq. (35). For Pauli noise, θ 2 = O(r 2 ), so we recover eq. (28). Conversely, for unitary noise (d − 1)θ 2 = 2dr + O(r 2 ), so we recover eq. (11) in such regime. Moreover, the above bound is saturated (to the appropriate order) in even dimensions by channels of the form These include single qubit amplitude damping and dephasing channels combined with a unitary evolution around the dampening/dephasing axis.
The following two theorems result from more general matrix inequalities that we prove in appendix A. We apply the inequalities to the unital block of the Liouville representation from eq. (9), and substitute the expressions for the RB decay rate and the unitarity from eq. (10). For theorem 4, we also use results from [12], which state that the maximal singular value of the unital block is upper-bounded by d 2 for general channels and 1 for unital channels.
Theorem 3. For any two quantum channels X and Y, Theorem 4. For any m channels X i with p(X i ) = p, u(X i ) = u, the RB decay rate of the composite channel satisfies Furthermore, if the X i are unital channels, the bound can be improved to

V. APPLICATION: INTERLEAVED RB
The fidelity extracted from standard RB experiments typically characterizes the average error over a gate set G, defined as However, one might only care about the fidelity F (E h ) attached to a specific gate of interest h ∈ G, such as one of the generators required for universal quantum computing. The interleaved RB protocol [6] yields a fidelity estimate of E h E, the composition between the single gate error and the gate set error, which provides bounds on the desired value F (E h ). An issue with this approach is that these bounds generally have a wide range, since possible coherent effects cannot be ignored. This issue is illustrated by the results of two simulations of interleaved RB experiments, plotted in fig. 1. In both scenarios, the fidelity of the gate error and of the composed gate were fixed at F (E) = 0.9975 and F (E h E) = 0.9960 respectively, hence leading to the same single gate fidelity estimate. In the first case, the interleaved gate h is unitary with high fidelity (F (E h ) = 0.9991), whereas in the second case the error is depolarizing, with a lower fidelity (F (E h ) = 0.9975). This example illustrates how interleaved RB, without a measure of unitarity, can only provide a loose estimate of the infidelity of an individual gate. More generally, rearranging the bound in theorem 1 to isolate χ Y 00 gives Moreover, this bound cannot be improved without further information. Now suppose that r(E h E) ≈ 2r(E), so that the uncertainty of r(E h ), obtained via eq. (37) and table I is While this bound does give an estimate of the infidelity, this estimate is comparable to the following naive estimate that requires no additional experiment. As the fidelity, and hence the infidelity, is a linear function of E we have which, since r(E) is nonnegative for any channel E, implies for any h ∈ G. (Note also that this bound can be heuristically improved by identifying sets of gates that are expected to have comparable error.) When G is chosen to be the 12-element subgroup of the Clifford group that forms a unitary 2-design, the naive bound is, at the very worst, a factor of 3/ √ 2 worse than the bound from interleaved benchmarking and requires no additional statistical analysis or data collection.
However, if the error channels were guaranteed to be depolarizing, F (E h ) could be exactly estimated from an  interleaved RB experiment. In general, we can use the coherence angle to quantify how close the noise is to depolarizing noise. From theorem 3, we then have the following bounds, which can be orders of magnitude tighter as illustrated in fig. 2.
Corollary 5. For any two quantum channels X and Y, where γ xy := p(X Y)/ u(Y).

VI. SUMMARY AND OUTLOOK
We have proven that a coherent composition of unitary quantum channels results in the fastest decay of fidelity. In this case, the error rate grows quadratically in the number of gates, in contrast with the linear growth for stochastic Pauli channels.
The disparity between these two regimes prevents an accurate fidelity prognosis if only the individual gate fidelity is known. Hence, in order to characterize more precisely the evolution of fidelity, we introduced a coherence angle-see eq. (29)-which enables a tighter bound on the total error in a quantum circuit in terms of efficiently estimable quantities that smoothly interpolates between the linear and quadratic regimes.
As an immediate application, we demonstrated that this bound substantially improves the estimates of individual gate fidelities from interleaved randomized benchmarking, which, in the absence of the improved bound, are comparable to the naive bound obtained by noting that the infidelity from standard RB is the average of the infidelities of the individual gates.
Moreover, both bounds are saturated for all values of M 1 F , M 2 F , θ(M 1 ), and θ(M 2 ) in even dimensions.
Proof. By the Cauchy-Schwarz inequality, Setting D i := Tr Mi d I d for i = 1, 2, S is maximized when p = 1, in which case it equals m 2 .