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A new class of efficient randomized benchmarking protocols

Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, Stephanie Wehner

arXiv:1806.02048v2quant-ph

TL;DR

For general gatesets, randomized-benchmarking data can be difficult to interpret robustly. Character randomized benchmarking uses representation theory to isolate decay channels so their quality parameters can be fitted one at a time. Favg(C) lower bound: ≈0.62 with standard interleaved benchmarking and ≈0.79 with 2-for-1 interleaved benchmarking.

  • Problem

    For general gatesets, randomized-benchmarking data can be difficult to interpret robustly.

    Source: Methods · pp. 5–6

  • Method

    Character randomized benchmarking uses representation theory to isolate decay channels so their quality parameters can be fitted one at a time.

    Source: Results · pp. 2–3

  • Results

    Favg(C) lower bound: ≈0.62 with standard interleaved benchmarking and ≈0.79 with 2-for-1 interleaved benchmarking.

    Source: Discussion · p. 5

  • Takeaways & Limitations

    The reported lower bound is ≈0.62 with standard interleaved randomized benchmarking, versus Favg(C) ≈0.79 guaranteed with 2-for-1 interleaved randomized benchmarking using fewer experimental resources.

    Source: Discussion · p. 5

    The general interleaved-character result assumes an irreducible mixing matrix, though the proof also extends to direct sums of irreducible matrices.

    Source: Methods · p. 10

Abstract

from arXiv · show

Randomized benchmarking is a technique for estimating the average fidelity of a set of quantum gates. For general gatesets, however, it is difficult to draw robust conclusions from the resulting data. Here we propose a new method based on representation theory that has little experimental overhead and applies to a broad class of benchmarking problems. As an example, we apply our method to a gateset that includes the $T$-gate, and analyze a new interleaved benchmarking protocol that extracts the average fidelity of a 2-qubit Clifford gate using only single-qubit Clifford gates as reference.

Results

Character randomized benchmarking uses representation theory to isolate decay parameters and recover average fidelity for a broad class of gatesets. The paper illustrates it with T-gate benchmarking and a 2-for-1 interleaved protocol.

Character randomized benchmarking

  • Character randomized benchmarking isolates individual decay channels for single-exponential fits, including for non-Clifford gatesets and in the presence of leakage.Estimating all quality parameters allows the gateset's average fidelity to be obtained.Source: Results · pp. 2, 4

Benchmarking T-gates

  • For the CNOT-dihedral group Tq, the protocol fits f2 and f3 separately to recover the average fidelity of a gateset that includes T gates.The Pauli group is used as the character group to isolate these parameters.Source: Discussion · p. 4

2-for-1 interleaved benchmarking

  • The 2-for-1 protocol performs interleaved benchmarking of a 2-qubit Clifford gate using single-qubit Clifford reference gates.This reduces experimental requirements; the authors also report using the protocol to characterize a CPHASE gate in spin qubits.Source: Discussion · pp. 4–5
  • The 2-for-1 protocol guarantees a lower bound of Favg(C) ≈0.79, versus ≈0.62 for standard interleaved benchmarking, with fewer experimental resources.The protocol also extracts a measure of correlation between the two qubits.Source: Discussion · p. 5

Further applications

  • Character benchmarking can estimate fidelities directly over groups formed by native gates, avoiding the need to compile the Clifford group from those gates.The authors note this advantage is especially pronounced when native two-qubit gates are non-Clifford, such as SWAP.Source: Discussion · p. 5

Methods

This section analyzes the resource costs and noise behavior of character randomized benchmarking, including conditions under which interleaved benchmarking yields an exponential decay. It uses mixing-matrix powers and perturbation theory to bound deviations from single-exponential behavior, with a CPHASE interleaved-benchmarking example.

Scalability of character randomized benchmarking

  • Scalable character randomized benchmarking requires polynomially many quality parameters and polynomial growth in the projected character dimension as qubit count increases.The resource cost depends on the number of irreducible subrepresentations and the character-function magnitude; not every benchmarking-group family is scalable.Source: Methods · p. 6
  • For examples with the Pauli character group, standard statistical techniques yield a 99% confidence interval of size 0.02 around kλ′; estimating m(⃗G) takes 1,769 repetitions, independent of qubit count q.The maximal character value in these examples is 1.Source: Methods · p. 6
  • The polynomial sequence-count bound may be overly pessimistic; few-hundred-sequence estimates are known for Clifford randomized benchmarking, while estimates for other families remain open.The authors suggest such results could partly generalize to other benchmarking groups and adapt to character randomized benchmarking.Source: Methods · p. 6

Gate-dependent noise

  • Under gate-dependent noise, character randomized benchmarking data fit a single exponential up to a small, exponentially decreasing correction.The model assigns a channel to each gate but excludes time-dependent and memory effects; the decay rate is the largest eigenvalue of an averaged operator.Source: Methods · p. 7
  • The resulting decay rates characterize average noise between gates, extending the standard randomized-benchmarking interpretation to character benchmarking.The rates relate to the average fidelity of noise between gates, averaged over the gates.Source: Methods · pp. 7, 9

Interleaved character randomized benchmarking

  • Interleaved character benchmarking yields an exponential fit under an irreducible mixing-matrix condition, which the authors verify for 2-for-1 benchmarking with CPHASE.For the theorem’s parameters, δ1 = O(1 − Favg(ECE)) and δ2 = γ + O([1 − Favg(ECE)]^2), where γ is the second-largest eigenvalue magnitude of the mixing matrix.Source: Methods · pp. 9–10

Methods

  • Under irreducibility of M(I), Perron-Frobenius gives a non-degenerate largest-modulus eigenvalue; perturbing ECE=I−αF yields γmax(M(ECE))=1−α+O(α²).The analysis also uses perturbation theory to show that the sequence vector has high overlap with the dominant right eigenvector.Source: Methods · pp. 10–11
  • When ECE=I, the sequence expression reduces to the expected ideal value.The passage identifies that value as ⟨⟨Q|P̂φ|ρ⟩⟩.Source: Methods · p. 11
  • At high fidelity α is small, and the second-largest eigenvalue γ differs from that of M(I) by O(α²); the error terms scale as δ1=O(1−Favg(ECE)) and δ2=|γ|+O((1−Favg(ECE))²).These conditions support the perturbative analysis of the dominant matrix-power contribution.Source: Methods · p. 11
  • 1, 1/3, and −1/9 are the eigenvalues of the CPHASE mixing matrix; deviations from a single exponential are at most (1/3)^m at sufficiently high fidelity, negligible even for fairly small m.The example uses two copies of single-qubit Clifford gates. A similar result was obtained by different methods [49].Source: Methods · pp. 11–12

I. Data Availability Statement

The only data and analysis generated for the work are those used for Supplementary fig. 2, which will be available online.

I. Data Availability Statement

  • Data and analysis used for Supplementary fig. 2 will be available online; no other supporting data were generated or analyzed.The passage gives the DOI https://doi.org/10.5281/zenodo.2549368.Source: §I · p. 12

A. Supplementary Methods I: Background material

This supplementary section reviews representation theory and the Pauli transfer matrix formalism for quantum channels.

A. Supplementary Methods I: Background material

  • The supplementary methods review representations, characters, and the Pauli transfer matrix formalism for quantum channels.The section points readers to for representation and character background and for its quantum-channel presentation.Source: A. Supplementary Methods I: Background… · p. 14

1. I.1 Representation theory

This section introduces finite-group representations and their decomposition into irreducible subrepresentations, then recalls characters and projection tools used in the analysis.

1. I.1 Representation theory

  • A representation maps a finite group G to unitary linear transformations on a finite-dimensional vector space V.The text defines reducibility through an invariant nontrivial subspace and calls the restricted action a subrepresentation.Source: §1 · p. 14
  • Two representations are equivalent when an invertible linear map relates their actions, while twirling averages a linear map with respect to a representation.The twirl is denoted Tφ.Source: §1 · p. 14
  • Maschke’s lemma decomposes representations into irreducible components, and the paper mostly treats multiplicity-free representations with inequivalent components.In a multiplicity-free representation, each irreducible component occurs once.Source: §1 · p. 14
  • A representation’s character is defined as a function on the group, and the paper restricts attention to real-valued characters.The text recalls character properties for pairs of representations and their direct sums and tensor products.Source: §1 · p. 15
  • The generalized projection formula uses a character to project onto the support of subrepresentations equivalent to a chosen irreducible representation.When equivalent copies occur, the projector includes all such subrepresentations.Source: §1 · p. 15

2. I.2 Pauli transfer matrix representation of quantum channels

The Pauli transfer matrix (PTM) represents quantum channels as matrices acting on vectors in a basis of normalized Pauli matrices. It preserves channel composition and tensor products as matrix operations.

2. I.2 Pauli transfer matrix representation of quantum channels

  • A channel acts on the vectorized density matrix by standard matrix multiplication in the PTM representation.The representation is built from a basis of normalized Pauli matrices spanning the Hermitian-matrix space.Source: §2 · pp. 15–16
  • Composing channels and taking their tensor products correspond to multiplying and tensoring their PTM matrices, respectively.The PTM representation of a unitary subgroup is also an actual representation.Source: §2 · p. 16

B. Supplementary Methods II: Standard randomized benchmarking with arbitrary finite groups

Standard randomized benchmarking samples finite-group gate sequences, applies an inversion, and averages survival probabilities over sequences and lengths. Under the stated multiplicity-free representation condition, the analysis defines quality parameters for the group’s implemented gates.

B. Supplementary Methods II: Standard randomized benchmarking with arbitrary finite groups

  • The stated analysis assumes gate-independent noise and a finite group whose PTM representation is multiplicity-free.The lemma concerns the average survival probability for an input state and two-component POVM.Source: B. Supplementary Methods II: Standard… · p. 16
  • The protocol samples sequences of length m, applies the inverse of the ideal sequence, and averages measured survival probabilities across sequences and lengths.It begins with a state and two-outcome POVM chosen so Tr(Qρ) is large.Source: B. Supplementary Methods II: Standard… · p. 16
  • Each quality parameter fλ is defined as Tr(PλE)/Tr(Pλ), with Pλ the projection associated with a representation sector.The proof states this definition after applying Schur’s lemma.Source: B. Supplementary Methods II: Standard… · p. 17

C. Supplementary Methods III: Average fidelity and quality parameters

The section relates average fidelity to the quality parameters obtained in character randomized benchmarking. Under a multiplicity-free PTM representation, average fidelity is related to a weighted average of those parameters.

C. Supplementary Methods III: Average fidelity and quality parameters

  • The average fidelity of a channel is defined by averaging its output-state overlap over pure states with respect to Haar measure.The section also relates average fidelity to the trace of the channel’s Pauli transfer matrix.Source: C. Supplementary Methods III: Average… · p. 17
  • Average fidelity Favg(E) is related to a weighted average of the quality parameters fλ generated by character randomized benchmarking.The relation is stated for a group with a multiplicity-free PTM representation.Source: C. Supplementary Methods III: Average… · p. 18

D. Supplementary Methods IV: Character randomized benchmarking

Character randomized benchmarking uses representation sectors and weighted survival probabilities to estimate quality parameters from randomized gate sequences. Its interleaved version inserts a target gate into the sequence and specifies a corresponding inverse and weighting procedure.

D. Supplementary Methods IV: Character randomized benchmarking

  • Character randomized benchmarking requires a group with a multiplicity-free PTM representation and outputs weighted survival-probability data for chosen sequence lengths.A state and two-component POVM are chosen to maximize the contribution from a selected representation sector.Source: D. Supplementary Methods IV: Character… · p. 18
  • The protocol samples group gates, applies their inverse after the sequence, and weights the measured survival probability by a subgroup character.The selected representation and state-POVM choice determine the sector being measured.Source: D. Supplementary Methods IV: Character… · p. 19
  • Averaged data across sequences and lengths are fitted to an exponential decay to extract the quality parameter fλ′.The resulting quality parameters can be associated with average fidelity using lemmas 4 and 5.Source: D. Supplementary Methods IV: Character… · p. 19
  • The interleaved protocol inserts target gate C between sampled gates and requires the group generated by G and C to be finite.It compiles the initial sampled gate and subgroup gate together, does not invert the subgroup gate, and does invert C.Source: D. Supplementary Methods IV: Character… · p. 19
  • For interleaved benchmarking, the protocol samples the subgroup gate, weights survival probability by its character, and averages over subgroup choices and gate sequences.The output is a sequence of estimates indexed by the chosen lengths.Source: D. Supplementary Methods IV: Character… · p. 19

E. Supplementary Methods V: Examples of character randomized benchmarking

The examples section uses the multi-qubit Pauli group to construct character randomized benchmarking channels. Its Pauli transfer matrix representation decomposes into inequivalent one-dimensional representations whose characters can be computed efficiently.

1. V.1 Representations of the Pauli group

  • The Pauli group’s PTM representation decomposes into one-dimensional, mutually inequivalent representations, one for each normalized Pauli matrix.The method uses these representations as distinct channels for character randomized benchmarking.Source: §1 · pp. 19–20
  • Each normalized Pauli matrix spans an irreducible subrepresentation, with its projector given by the corresponding rank-one projection.The character function for each representation is determined by the Pauli group’s action on that matrix.Source: §1 · p. 20
  • Pauli-group character values can be computed efficiently by checking whether the sampled Pauli commutes or anticommutes with the selected Pauli matrix.The calculation can happen in post-processing, so high-speed character evaluation is not required during the experiment.Source: §1 · p. 20

2. V.2 Benchmarking a T gate

For benchmarking a gateset containing T gates, the section uses the CNOT-dihedral group and its Pauli transfer matrix representation. Under CPTP noise, one quality parameter is fixed, leaving two parameters to estimate with character randomized benchmarking.

2. V.2 Benchmarking a T gate

  • The CNOT-dihedral gateset Tq consists of gates synthesized from T, X, and CNOT gates.CNOT gates may act between distinct qubits, while T and X gates act on individual qubits.Source: §2 · p. 20
  • The PTM representation of Tq has three irreducible subrepresentations, so estimating the average fidelity requires three quality parameters.The section identifies these as f1, f2, and f3.Source: §2 · p. 21
  • Assuming the noisy gates are CPTP maps, f1 = 1, leaving f2 and f3 to estimate in two character randomized benchmarking experiments.For f2, the protocol uses the Z⊗q Pauli component and a specified state and measurement; for f3, it uses X⊗q.Source: §2 · p. 21
  • The f3 experiment uses the X⊗q component, and the estimated parameters are combined with representation traces to calculate average fidelity.The trace quantities used in the formula are reported as derived in.Source: §2 · p. 21

3. V.3 2-for-1 interleaved benchmarking

The 2-for-1 protocol benchmarks a two-qubit gate using two copies of the single-qubit Clifford group as the reference. In the simulation, it yields a higher guaranteed lower bound on the gate’s average fidelity than standard two-qubit interleaved benchmarking.

3. V.3 2-for-1 interleaved benchmarking

  • The protocol replaces the two-qubit Clifford reference group with two copies of the single-qubit Clifford group to benchmark a two-qubit gate C.It has separate reference and interleaved experiments.Source: §3 · pp. 21–22
  • The reference stage estimates the quality parameters for sectors (1,0), (0,1), and (1,1), using f(0,0) = 1 under CPTP noise to obtain the reference fidelity.The resulting parameters enter the stated expression for the average reference fidelity.Source: §3 · p. 23
  • The interleaved stage repeats character benchmarking with C inserted and uses the corresponding sector parameters to obtain the interleaved fidelity.The protocol specifies experiments for the (1,0), (0,1), and (1,1) sectors.Source: §3 · p. 23
  • The conventional interleaved gate fidelity estimate can be calculated, but without further knowledge of the noise process it is not a point estimate of the gate’s average fidelity.The estimate is only a point in the interval obtained by solving the stated bounds.Source: §3 · p. 24
  • Higher reference fidelity gives sharper bounds, and the section expects 2-for-1 benchmarking to have a higher reference fidelity when single-qubit gates are more faithful than two-qubit gates.The simulation uses single-qubit average fidelity 0.99 and two-qubit average fidelity 0.898; it reports reference fidelities of ≈0.98 for 2-for-1 and ≈0.87 for standard benchmarking.Source: §3 · p. 24
  • The guaranteed lower bound on average gate fidelity is ≳0.79 for 2-for-1 benchmarking, versus ≳0.62 for standard interleaved benchmarking.The comparison is from the simulated experiment.Source: §3 · p. 24

F. Supplementary Methods VI: Finite sampling

The finite-sampling discussion explains why character randomized benchmarking relies on empirical estimates of means and confidence intervals. It contrasts exhaustive estimation over the character group with a direct sampling procedure for estimating the character average.

F. Supplementary Methods VI: Finite sampling

  • Finite samples require empirical estimates of probability-distribution means, with confidence intervals used to express their reliability.The section introduces Hoeffding’s inequality for bounded distributions to construct such intervals.Source: F. Supplementary Methods VI: Finite… · p. 24
  • Each measurement outcome is a Bernoulli random variable whose mean is the survival probability for the associated quantum operation.The discussion separates randomness from sequence sampling, character-group sampling, and quantum measurement.Source: F. Supplementary Methods VI: Finite… · p. 25
  • Estimating the character average by separately measuring every character-group element requires |Ghat|N samples, which can grow exponentially with qubit count.For the Pauli group on q qubits, the group size is 4^q.Source: F. Supplementary Methods VI: Finite… · pp. 25–26
  • Instead, the protocol can estimate the character average directly by sampling a group element, preparing and measuring once, and weighting the outcome by its character value.The weighted sample is then used to form an empirical estimate.Source: F. Supplementary Methods VI: Finite… · p. 26

4. Repeat sufficiently many times and compute the empirical average of x( ˆG)

Repeated samples from a mixture distribution estimate the target mean. Bounds on the distribution make the per-parameter sample requirement independent of group size, and high-rank input states do not require additional samples.

4. Repeat sufficiently many times and compute the empirical average of x( ˆG)

  • The estimate is obtained from samples of a mixture distribution whose mean is the weighted average of the component means.The procedure samples a Bernoulli distribution for each selected group element and averages the resulting x( ˆG) values.Source: §4 · pp. 26–27
  • The sample requirement for estimating the mean does not depend on |G|, and the stated sample count is independent of qubit number q.The distribution bounds support applying a concentration inequality without a dependence on |G|.Source: §4 · pp. 26–27
  • For Pauli-group characters, the distribution is bounded by ±1; the example uses ε = 0.02 and confidence δ = 0.99.The one-dimensional character has magnitude 1 for every Pauli-group element.Source: §4 · pp. 26–27
  • For high-rank input states, sampling from a mixture of Bernoulli distributions still estimates the target mean without increasing the required number of samples.The required count depends on the interval on which Λλ is defined.Source: §4 · p. 27
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