Bounding quantum gate error rate based on reported average fidelity

Remarkable experimental advances in quantum computing are exemplified by recent announcements of impressive average gate fidelities exceeding 99.9% for single-qubit gates and 99% for two-qubit gates. Although these high numbers engender optimism that fault-tolerant quantum computing is within reach, the connection of average gate fidelity with fault-tolerance requirements is not direct. Here we use reported average gate fidelity to determine an upper bound on the quantum-gate error rate, which is the appropriate metric for assessing progress towards fault-tolerant quantum computation, and we demonstrate that this bound is asymptotically tight for general noise. Although this bound is unlikely to be saturated by experimental noise, we demonstrate using explicit examples that the bound indicates a realistic deviation between the true error rate and the reported average fidelity. We introduce the Pauli distance as a measure of this deviation, and we show that knowledge of the Pauli distance enables tighter estimates of the error rate of quantum gates.

Unitary quantum computing (UQC) could outperform classical computing for certain computational problems in the sense that resource costs such as time or space scale better than for the best-known classical algorithm [1].Famous examples include the provable quadratic speedup for search [2], presumed exponential speedup for the Abelian Hidden Subgroup Problem [3,4], and the contentious speedup for stoquastic Hamiltonians using adiabatic quantum computing [5].If a problem instance with an ℓ-bit input is solved within bounded error ǫ by an algorithm employing n bits or qubits (space cost) and ν Boolean or unitary gates (time cost), the algorithm is considered efficient if n, ν ∈ poly(ℓ) [6] and n, ν ∈ polylog(1/ǫ) [7].
In practice, preparation, processing and measurement are faulty, but the Threshold Theorem for Fault-Tolerant Quantum Computation ("Threshold Theorem") [8][9][10][11][12] guarantees the existence of a threshold success rate τ (0 < τ < 1) such that a faulty computer whose worstcase error rate ℘ satisfies ℘ < 1 − τ can perform universal quantum computations efficiently, namely with additional overhead poly(n, ν).The Threshold Theorem is crucial for establishing the viability of quantum computers as superior to classical computers.
A key drawback of the Threshold Theorem is that τ is established existentially, not constructively [11]; consequently, this scalability figure of merit is elusive in practice.An upper bound τ ub for τ is determined in practice by constructing a quantum error correcting code and then estimating the code-specific threshold [13].If the worst-case error rate ℘ of the faulty components satisfies the threshold ℘ < 1 − τ ub , concatenation of the code reduces the error to within ǫ with only poly(n, ν) overhead; the C4/C6 code, for example, yields τ ub ≈ 97% [14].
Current experimental characterizations of quantum gates do not report ℘.Instead the gate fidelity Φ [15] is the typical figure of merit for gate performance.Recent reports of Φ exceeding 99% for two-qubit gates [16] generate strong optimism about the feasibility of scalable quantum computing, yet Φ is not the correct quantity to assess scalability.Our aim is to convert experimentally reported Φ to an upper bound for ℘.This upper bound ℘ ub provides a sufficient condition for faulttolerant quantum computing: errors can be efficiently corrected if In order to derive ℘ ub , we first introduce the classical foundation of quantum gate error rates.In particular, we introduce the classical gate fidelity Φ (using˜notation to distinguish classical from quantum cases) and elucidate the distinction between Φ and ℘ that is present even for reversible classical computing (RCC).Whereas the RCC version Φ-℘ disconnect is mainly of academic interest, the UQC Φ-℘ disconnect has important practical consequences due to the popularity of Φ as an experimentally convenient figure of merit [17][18][19].
The disconnect between Φ and ℘ has been quantified by means of a pair of inequalities [20], which we use to derive ℘ ub .Our upper-bound scales as √ 1 − Φ, which we show to be optimal.We find that the bound is overly pessimistic if the quantum noise is well-approximated by a Pauli channel, though we emphasize that this condition is both theoretically and experimentally difficult to ensure.Finally, we evaluate some reported experimental results using ℘ ub and highlight the significance of this evaluation for future assessments of experimental quantum computing efforts.A quantum algorithm is implemented as a unitary operator U , which can be approximated by concatenating unitary gates drawn from a fixed finite-size universal gate set G .This set G must be able to generate a dense subset of SU (D) by concatenating elements.Specifically, given U and ǫ < 1, there exists some U ǫ for each universal gate-set G that satisfies and can be realized as a concatenation of polylog(1/ǫ) gates from G [7].We focus on qubits (d = 2), for which two important gates are the controlled-NOT CNOT |b 0 , b 1 := |b 0 , b 1 ⊕ b 0 and controlled-controlled-NOT (Toffoli) For qubits, G can comprise and CNOT [21] or, alternatively, H and CCNOT [22,23].
Experimental efforts directed at constructing scalable UQC aim to realize all elements of a certain G , and they characterize and report the error of each G ∈ G .Now we introduce the channel representation, which accounts for practical imperfections that cannot be expressed using the mathematics of pure states and unitary gates.A practical implementation of G cannot be represented by a unitary conjugation and is represented instead by a completely-positive, trace-preserving linear endomorphism, or "quantum channel", G acting on S (H s ).The practical implementation described by G is compared to the ideal unitary gate G ♮ (ρ) = GρG −1 for each input ρ ∈ S (H s ), by means of the "discrepancy channel" . For a perfect implementation, D G ≡ 1. Imperfections due to Markovian decoherence [24] and unitary approximation [7] may be assessed via the discrepancy channel model.
Imperfect RCC gate operations can be modelled analogously.In RCC, perfect inputs and outputs are equallength bit-strings s ∈ B n := {0, 1} n , and all operations act as permutations of inputs by Cayley's Theorem [25,26].Any permutation of possible input states may be decomposed into a product of CCNOT gates, provided that the gates are allowed to access an ancillary register that provides constant 0s and 1s as needed [25].Thus the gate set G = { CCNOT} is universal for RCC.
Imperfect RCC operations result in probabilistic outcomes: the outcome is a probability mass function serving as the state of the output string, namely ρ = (ρ s∈B n ) represented as a vector of length D = 2 n .Entries of ρ are elements of the real-number interval I := [0, 1], and the space of valid RCC states { ρ} is S(B n ) of probability mass functions over the set B n .Classical pure input states take the form δ s,s0 ∈ S (B n ) for some string s 0 , and the ideal RCC gate G is represented by G♮ acting on the input state as a permutation matrix.
An imperfect implementation of G is representable by a D × D doubly-stochastic matrix G, which is a classical version of a Markovian channel.From the Birkhoffvon Neumann Theorem, G is a convex linear combination of permutation matrices [27].The RCC analogue of the discrepancy channel is therefore the doubly-stochastic matrix D The discrepancy channel, in both classical and quantum cases, is a model of the difference between the ideal and the actual implementation.We wish to evaluate the error rate of this discrepancy channel, but the error rate depends on the input.Therefore, the global error rate depends on the distribution of input states, and examples include average-and worst-case error rate.For the error rate to be useful, it must be compared to a rigorously established threshold.
We define the error rate of a gate for a specific input by the probability that the gate produces the wrong output.The ideal output is G♮ (δ s,s0 ) = δ s,G(s0) , and the actual output is ρ = G (δ s,s0 ).The worst-case error rate is the maximum difference between the event probabilities given by these two different distributions, which is known as the total-variation distance d tv [28].For two distributions ρ and ρ′ , d tv (ρ, ρ′ ) = 1  2 |ρ − ρ′ | 1 , where | • | 1 represents the vector 1-norm.For example, if ρ0 is a pure state (1, 0, 0, 0) T (for T the transpose) that is to be compared to an impure state ρ = (1, 1, 1, 1) T /4, d tv (ρ, ρ0 ) = 3/4, which is the probability that the actual distribution ρ assigns a different output string from that produced by ρ0 .
In general, for a given input s 0 , the total-variation distance between the ideal and actual output distributions is (5) The worst-case RCC gate error rate is the maximum of this quantity over all inputs s 0 , namely  6) holds because • 1 is invariant under the action of metric-preserving basis transformations such as that executed by the permutation matrix G −1 ♮ .For example, consider G = CCNOT implemented such that its failure probability is p; i.e. the gate executes correctly with probability 1 − p and returns the input with probability p.The discrepancy channel can thus be expressed as the doubly-stochastic matrix D G = (1 − p) 1 + p G♮ , where G♮ is the 8 × 8 permutation matrix corresponding to a perfect implementation of G. From CCNOT(110) = 111 and 1(110) = 110, we obtain G♮ − 1 1 = 2 and therefore ℘ = p.Our definition of ℘ thus quantifies the failure probability as desired.
In contrast to the well motivated ℘ figure of merit, the classical analogue of Φ employs the Bhattacharyya coefficient, also known as the Hellinger affinity [28], which is the classical version of quantum state fidelity [29,30].
The Bhattacharyya coefficient is φ(p, q) := k p(k)q(k) with p, q probability distributions over the same finite domain of possible outcomes indexed by k.Thus, φ quantifies the overlap of two probability distributions and is one, rather than zero, for two identical distributions.Hence, φ is not a metric though it is closely related to a metric called the Hellinger distance [28,31].The Hellinger distance is in fact the classical version of the Bures metric [29] and is equivalent to the totalvariation distance in the sense that two distributions that are close as measured by the total-variation distance are also close as measured by the Hellinger distance and vice versa [31].Though the two metrics generally differ quantitatively, they agree qualitatively.
We are now prepared to define the classical analogue of gate fidelity.This fidelity is which does not seem to have warranted attention in the literature.As Φ has not previously been studied, the classical Φ-℘ connection remains unexplored.To illustrate the Φ-℘ connection, we calculate Φ for our previous example of a noisy implementation of CCNOT.We find φ G (δ s,s0 ) , G♮ (δ s,s0 ) = √ 1 − p, if s 0 = 110 or 111, 1, otherwise.(8) Therefore, we find the connection Φ = (3 + √ 1 − ℘)/4.Now that we have investigated the Φ-℘ connection in the classical setting, we are now equipped to explore the Φ-℘ relationship in the quantum setting.In contrast with the definition of Φ as an average over finitely many pure input states, the UQC gate fidelity Φ is defined as an average over a continuous collection of possible pure input states [15]: for dµ ψ representing the uniform probability measure parametrized by the pure-state label ψ.This integral can be supplanted by a unitary design [32,33], which is exploited for efficient experimental estimation of Φ [17].
Now we construct a quantum version of statedependent gate error analogous to Eq. ( 5).The distance between density operators ρ and ρ ′ is [30]  .Now that we have the state-dependent error rate, we construct the worst-case error rate for G.This construction cannot involve the operator-norm because of the failure of the important condition A⊗B ≡ A B , which is important because the error rate of a UQC gate can surpass the value assigned by the operator-norm induced by the trace-norm if the gate input is entangled with another part of the register [34].Any gate would be expected to process entangled inputs frequently during the normal operation of any UQC device.
To circumvent the failure of the operator-norm construction, the error rate associated with a UQC gate is instead quantified using the diamond norm • ⋄ [35], which is a stabilized version of • 1 that properly generalizes the definition of error rate from Eq. ( 6) [11]: We therefore define the error rate of an imperfect quantum gate by This definition of gate error rate has been related to the gate fidelity Φ by the expression [20] 1 and suggests our upper bound thus connect Φ to some threshold error rate 1 − τ ub : We do not know if ℘ ub is a tight bound for ℘, but we are certain that ℘ ub scales optimally with respect to each of the variables D and Φ if the other is fixed.We demonstrate this by considering a particular imperfect implementation of a gate G defined by G |s = |s for Our implementation is characterized by the discrepancy channel D G = (1 − λ)1 + λG ♮ for some value 0 < λ < 1.
We first compute Φ, then calculate ℘ ub , and finally compare ℘ ub to the true value of ℘.We use the well known formula [15] where {K ℓ } is any Kraus representation of D G , to compute Φ.
We employ the Kraus representation by a useful theorem for the diamond-norm distance between a unitary channel and the identity channel [36].The bound ℘ ub is saturated only in the limits λ → 0 or θ → 0.
We now address the scaling of ℘ ub by considering the limits λ → 1 and θ → π.Any upper bound denoted ℘ ′ ub must satisfy ℘ ′ ub ≥ ℘, from which we determine Thus the optimal choice of ℘ ′ ub scales as √ 1 − Φ for a fixed value of D and as D for a fixed value of Φ.Our bound ℘ ub therefore cannot be significantly tightened without additional promises about the discrepancy channel.
We obtain a potentially tighter bound if the discrepancy channel is promised to be close to its Pauli twirl DG (•) := 1/D 2 Ξ Ξ D G (Ξ • Ξ)Ξ, with the sum over all D × D Pauli operators Ξ.The resulting channel DG is Pauli.The lower bound of Expression (12) , where Φ is the fidelity of the channel DG .In fact, Φ = Φ because Φ is linear and invariant under unitary conjugation.Thus In seeking a tighter bound, we introduce for ∆ := 1 2 D G − DG ⋄ , which is the probability that the discrepancy channel D G acts differently from its Pauli twirl DG .We use the triangle inequality D G − 1 ⋄ ≤ D G − DG ⋄ + DG −1 ⋄ to establish that ℘ub is a tighter upper bound than ℘ ub , provided that ∆ is small.If an efficient measurement procedure that provides an upper bound on ∆ were known, we could provide tighter estimates of progress towards practical fault-tolerant quantum computation.
We demonstrate the use of our bound by comparing the performance of photonic integrated circuits to the C4/C6 code threshold τ ub = 97% [14].Fault-tolerant quantum computing is possible in this case with singlequbit gates satisfying Φ > 99.985% and two-qubit gates satisfying Φ > 99.9955% according to Expression (14).A recent theoretical proposal [38] for high-quality photonic integrated circuits claims a median value of Φ = 0.9977 for CNOT gates is achievable.From Eq. ( 13), we obtain that Φ = 0.9977 corresponds to ℘ ub ≈ 21.4%, which fails the threshold condition ℘ ub < 1 − τ ub ≈ 3% of Expression (1).
Now we analyze recent results in another promising architecture: superconducting quantum circuits with surface codes.The surface code threshold is reported to be τ ub = 99.43%[39], which corresponds to a singlequbit fidelity threshold of 99.99946% and a two-qubit fidelity threshold of 99.99984% by Expression (14).Therefore, as the surface code comprises mostly two-qubit gates [39], a target of Φ > is appropriate to reach the requisite τ ub = 99.43%rather than the current standard of Φ > 99% [16].Current experiments surpass Φ = 99.9% for single-qubit gates and Φ = 99% for two-qubit gates [16], which falls short of our recommended target.In terms of the gate error rate, these reported values yield ℘ ub < 7.7% for single-qubit gates and ℘ ub < 44.7% for two-qubit gates, which fail the threshold condition (1): ℘ ub < 1 − τ ub ≈ 0.57%.
Reports of extremely high gate fidelities engender optimism that current technology is near the threshold required for fault-tolerant quantum computation.Yet, although the gate fidelity is an experimentally convenient figure of merit, it is not the proper metric, i.e., the worstcase quantum gate error rate, for assessing progress towards fault-tolerance.We have provided a sobering assessment of the current status of quantum computing technology by converting the gate fidelity to the worstcase quantum gate-error rate.Based on the best theoretical results currently available, we have shown that two-qubit gates must surpass 99.9955% gate fidelity to meet the generous C4/C6 fault-tolerance threshold.
We appreciate valuable discussions with C. Granade, P. Groszkowski, M. Mosca, and J. Watrous.YRS acknowledges financial support from the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the U.S. Army Research Office.JJW acknowledges financial support from the U.S. Army Research Office through grant W911NF-14-1-0103.BCS acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada, Alberta Innovates Technology Futures, and China's 1000 Talents Plan.All statements of fact, opinion, or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the U.S. Government.

with • 1
representing the operator-norm induced by | • | 1 .The far right-hand side of Eq. ( ) with E a of positive operator valued measure (POVM), and • Tr = Tr(| • |), with Tr denoting the trace of the absolute value of a square matrix inserted in place of •.Thus, the s-dependent error rate for G is d op (G(|s s|), G ♮ (|s s|)) To establish the classical foundation of error rates, we first describe ideal quantum computing involving pure states and unitary gates.A valid quantum state is ρ ∈ S (H s ) for H s the system Hilbert space (e.g.⊗ n ı=1 H