Theory of quantum system certification: a tutorial

I. INTRODUCTION

• Quantum devices promise applications from digital computing and analogue simulation to cryptographically secure communication, but their complexity makes correct-function certification difficult with classical simulation.This complexity motivates a broad landscape of protocols with differing assumptions, resource requirements, and information gain.Source: §I · p. 2

A. Anatomy of quantum certification protocols

• Certification protocols can be organized by assumptions, resource complexity, and information gain, balancing discriminatory power against measurement, sampling, and classical-processing costs.Protocols may involve a device, measurement apparatus, and classical processor; certification is a focused form of quantum-system characterization, while benchmarking compares devices through reproducible performance measures.Source: A. Anatomy of quantum certification… · pp. 2–3

B. Quantum certification for near-term devices – scalable certification of the physical-layer

• The tutorial targets noisy, intermediate-scale quantum devices whose physical layers require scalable certification because full tomography is infeasible, while physical-layer diagnostics remain useful even for future higher-level systems.It models devices using quantum states and processes, emphasizes digital quantum computing, and excludes application-level verification and property-specific self-testing; randomized benchmarking and cross-entropy benchmarking are tailored to digital devices.Source: B. Quantum certification for near-term… · p. 4

C. Overview and structure

• The tutorial combines protocol presentations with mathematical preliminaries and rigorous performance guarantees, covering state certification in Section II and process certification in Section III.Its state topics include fidelity-based certification, direct and shadow fidelity estimation, while process topics include direct process certification, randomized benchmarking, and cross-entropy benchmarking.Source: B. Quantum certification for near-term… · p. 4, C. Overview and structure · pp. 4–5

II. QUANTUM STATES

• The first part develops quantum-state certification from mathematical notation through statistical testing, tail bounds, state metrics, fidelity protocols, importance sampling, and shadow fidelity estimation.The mathematical and protocol chapters alternate between foundations and applications, with further reading offered for protocol variants and extensions.Source: C. Overview and structure · p. 5, §II · p. 5

A. Mathematical objects of quantum mechanics

• Finite-dimensional quantum mechanics represents systems on complex Hilbert spaces, states as trace-one positive semidefinite density operators, and discrete measurements as POVMs whose effects induce outcome distributions.Pure states are rank-one density operators of the form ρ = |ψ⟩⟨ψ|, while general states are convex combinations; the formalism introduces operators, adjoints, bra-ket notation, random variables, and traces.Source: A. Mathematical objects of quantum… · pp. 5–6
• Mixed quantum states are convex combinations of pure states with a normalized, entry-wise non-negative probability vector.A mixed state is any quantum state that is not pure.Source: A. Mathematical objects of quantum… · p. 6
• Composite quantum systems are represented by tensor-product Hilbert spaces, with H1 ⊗ H2 describing systems having spaces H1 and H2.The tensor-product construction embeds operators on one subsystem into the joint operator space.Source: A. Mathematical objects of quantum… · pp. 6–7
• A subsystem’s reduced state is obtained by taking the partial trace over the other subsystem, retaining all information accessible through measurements on that subsystem alone.For a bipartite state ρ, the reduced state is ρ1 := Tr2[ρ].Source: A. Mathematical objects of quantum… · p. 7
• The swap operator exchanges the two factors of a bipartite state, mapping a coefficient matrix A to its transpose A⊺.This construction is introduced alongside the swap-trick exercise.Source: A. Mathematical objects of quantum… · p. 7

B. A definition of quantum state certification

• A certification test measures multiple copies of a prepared state and accepts or rejects whether it approximates target state ρ within distance ϵ.The protocol’s output is based on measurement data and classical post-processing, with Figure 2 depicting rejection when dist(ρ, ˜ρ) > ϵ.Source: B. A definition of quantum state… · p. 7, Figure 2 · p. 7
• Completeness requires likely acceptance of the target state, while soundness requires likely rejection when the prepared state exceeds the tolerated distance ϵ.In the hypothesis-testing interpretation, both type-I and type-II errors are bounded by 1/3.Source: B. A definition of quantum state… · pp. 7–8
• Certification guarantees commonly assume independent or iid. preparations, although de Finetti methods or improved analyses can remove iid. at higher measurement cost.Environmental drift can violate iid. behavior, while entangled preparations could challenge independence-based analyses.Source: B. A definition of quantum state… · p. 8
• Sampling complexity measures consumed copies as a function of d and ϵ, while measurement complexity concerns implementation demands such as simultaneous-copy requirements and POVM circuits.Sequential measurements avoid storing copies, and efficient digital protocols are often associated with sampling complexity O(polylog(d)).Source: B. A definition of quantum state… · p. 8
• Full tomography can require at least 2^n measurements for pure n-qubit states, whereas fidelity estimation and certification can use dramatically fewer measurements under suitable structure.Certification may further improve the ϵ-dependence of sample complexity compared with fidelity estimation.Source: B. A definition of quantum state… · pp. 8–9

C. Estimation and tail bounds

• Estimators are measurement-outcome random variables whose bias, consistency, finite-sample ϵ-accuracy, and confidence determine how many samples a protocol requires.Estimator sampling complexity can be derived from tail bounds on random variables.Source: C. Estimation and tail bounds · p. 9
• Markov, Chebyshev, and Hoeffding inequalities bound deviations under progressively stronger conditions, while median-of-means improves tails for unbounded iid. variables with finite variance.Bernstein can improve concentration for bounded variables with controlled variance, Azuma relaxes independence, and a union bound handles multiple events.Source: C. Estimation and tail bounds · p. 10

D. Expectation value estimation for observables

• Expectation values satisfy ⟨A⟩_ρ = Tr[ρA] and can be estimated from repeated projective measurements using an unbiased empirical mean.The observable’s outcomes are its eigenvalues, sampled according to the associated PVM and state ρ.Source: D. Expectation value estimation for… · p. 11
• Bounded observables can be estimated with dimension-independent measurement effort, while increasing the number of measurements improves confidence exponentially fast.Hoeffding’s inequality supplies the sampling-complexity bound for independent outcomes from bounded spectra.Source: D. Expectation value estimation for… · p. 11
• Expectation-value estimation induces ε-certification tests for state distances defined through sets of observables.The protocol therefore connects observable measurements with certification under such distance measures.Source: D. Expectation value estimation for… · p. 12

Further reading

• For multiple observables, a union bound gives the sampling complexity when each uses a separate measurement setting, whereas shadow estimation can reuse one setting for suitable observables [19] [20] [21].Shadow-estimation techniques are discussed later in the context of state certification.Source: Further reading · p. 12

E. Distance measures for quantum states

• Quantum-state distances are supported by operator norms, including the spectral, trace, and Frobenius norms, which are Schatten p-norms for p = ∞, 1, and 2.Schatten norms are defined from the singular-value spectrum and obey Hölder-type inequalities and norm ordering.Source: E. Distance measures for quantum states · p. 12
• Trace distance is the maximal distinguishability of two quantum states under dichotomic POVM measurements, attained by the projector onto the positive part of ρ − σ.It is also the largest Schatten norm among the standard choices and thus gives the most pessimistic distance measure.Source: E. Distance measures for quantum states · pp. 12–13
• Fidelity measures state closeness rather than distance, ranges from 0 to 1, equals 1 for identical states, and reduces to squared overlap for pure states.For pure states, F(|ψ⟩⟨ψ|, |φ⟩⟨φ|) = |⟨ψ|φ⟩|^2.Source: E. Distance measures for quantum states · p. 13
• The Fuchs–van-de-Graaf inequalities relate trace distance and fidelity without explicit dimension dependence, but their square-root scaling can make 10^-3 trace distance require 10^-6 infidelity for certification.The square-root scaling is unavoidable for pure states, and subsequent sections introduce direct and shadow fidelity-estimation protocols.Source: E. Distance measures for quantum states · pp. 13–14

Naive direct certification

• Projective certification of a pure target accepts a single-copy pass outcome, is perfectly complete, and uses repeated copies to detect infidelity deviations with confidence 1 −δ.The protocol has zero type-I error, while repeated measurements control false acceptance; its sample complexity is independent of physical system size.Source: F. Direct quantum state certification · p. 14

Feasible measurement strategies

• The practical direct protocol samples feasible binary POVMs according to a probability distribution, yielding an effective measurement operator that replaces the inaccessible target projector.The strategy must accept the target with probability one, requiring Tr[Mρ] = 1 for every feasible measurement element.Source: F. Direct quantum state certification · p. 15

Performance guarantees

• A feasible strategy achieves certification when its effective operator has a positive spectral gap, with sample complexity incurring an overhead factor 1/ν(Ω) over naive projection.The protocol remains perfectly complete and certifies infidelity deviations with confidence 1 −δ.Source: F. Direct quantum state certification · pp. 15–16
• When the effective operator has a constant spectral gap, feasible direct certification has the same sample-complexity scaling as measuring the target state itself.The worst-case false-acceptance probability is determined by the second-largest eigenvalue through the spectral gap.Source: F. Direct quantum state certification · pp. 16–17

Minimax optimization

• Minimax optimization selects the feasible strategy minimizing worst-case false acceptance, reducing to the target projector without measurement restrictions and supporting optimal strategies for stabilizer states.For stabilizer states, the section derives a minimax-optimal certification protocol using two-outcome Pauli measurements.Source: F. Direct quantum state certification · p. 17

Stabilizer states

• A stabilizer state is the unique joint eigenvalue-1 state of an Abelian subgroup of the n-qubit Pauli group, generated by n elements and containing 2^n elements.The subgroup cannot contain −1.Source: §1 · p. 17
• Uniformly measuring each nontrivial stabilizer observable with probability 1/(2^n−1) and accepting the +1 eigenspace is minimax optimal for Pauli measurements [24].The resulting operator fixes the target stabilizer state and has the second-largest eigenvalue specified by the theorem.Source: §1 · p. 17
• Restricting certification to Pauli measurements incurs at most a constant overhead of 2, while only randomly subsampled stabilizer observables are measured [24].Although the stabilizer group contains 2^n−1 nontrivial stabilizers, the protocol measures only a small random subset.Source: §1 · p. 18

Motivation

• Direct certification gives only an accept/reject answer, motivating protocols that explicitly estimate the distance or fidelity between an implementation ˜ρ and target ρ.Source: §2 · p. 18

Fidelity estimation

• Homogeneous effective measurements let the direct quantum state certification protocol estimate the fidelity F(˜ρ, ρ) from the expectation value of Ω.For stabilizer states, the homogeneity parameter is τ = 2n−1−1 [14] [26].Source: §2 · pp. 18–19

Sampling requirements

• Estimating the expectation value to additive error ε requires measurements scaling as 1/ε^2; if Ω is directly observable, m ≥ 1/(2ε^2) ln(2/δ) achieves confidence 1 −δ.Protocol 14 instead assumes measurements that reproduce Ω only on average.Source: §2 · p. 19
• When the measurement set |M| is exponentially large, as for stabilizer states, Monte Carlo estimation and importance sampling can make fidelity estimation efficient.These methods motivate the subsequent discussion of direct fidelity estimation.Source: §2 · p. 19

Further reading

• Direct certification has been extended to maximally entangled, stabilizer, bipartite, hypergraph, weighted graph, Dicke, and adversarial state-preparation settings, often with close-to-optimal sampling complexity.Related protocols also address graph-state verification, Hamiltonian ground states, fidelity witnesses, matrix-product and permutationally invariant states, and non-iid. preparations.Source: Further reading · p. 19

G. Importance sampling

• Importance sampling rewrites an expectation under p using samples from q, reducing variance when q approximates the optimal distribution, although the exact optimum requires solving the original integration problem.For nonnegative f, the optimal distribution has zero variance, but its normalization factor is itself equivalent to the integration problem; empirical variance estimates error and increasing m controls it.Source: G. Importance sampling · pp. 19–20

H. Direct fidelity estimation

• DFE estimates Tr[ρ˜ρ] for a pure target state by expanding it in Pauli products, sampling observables according to the target-dependent importance distribution, and estimating their expectation values from measurements.The resulting random variable is an unbiased fidelity estimator, and the procedure combines classical sampling of Pauli indices with repeated probabilistic measurements.Source: H. Direct fidelity estimation · pp. 20–21
• DFE achieves an 2ϵ-accurate unbiased fidelity estimate with confidence 1 −2δ for iid. preparations, while its sample complexity scales linearly in Hilbert-space dimension rather than tomography’s ˜Ω(d2 rank(˜ρ)2) Pauli-measurement scaling.The guarantee applies to the expected state-preparation cost stated in Theorem 22.Source: H. Direct fidelity estimation · p. 21
• DFE can be extended from Pauli observables to arbitrary orthonormal bases and over-complete frames, although larger operator norms can increase the sampling complexity.The section also describes converting a fidelity estimate into a trace-distance certification test through an acceptance threshold.Source: H. Direct fidelity estimation · pp. 22–23
• For well-conditioned pure target states, modified DFE gives an ϵ-accurate unbiased estimate with confidence 1 −δ and sampling complexity independent of system size.Stabilizer states are a prominent example, with well-conditioning parameter α = 1.Source: H. Direct fidelity estimation · pp. 22–23
• What sampling complexity does direct fidelity estimation achieve for well-conditioned states?Source: H. Direct fidelity estimation · p. 23
• How can the protocol become a robust (ϵ, ϵ′)-certification test that accepts states within an ϵ′-trace norm ball around ρ with confidence 1 −δ?Source: H. Direct fidelity estimation · p. 23

I. Random states and unitaries

• Haar-random unitary moments are useful in certification and estimation because random unitaries with a fixed measurement can rapidly reveal information about the entire state space.The Haar measure is the unique left- and right-invariant probability measure on the compact unitary group U(d).Source: §I · p. 23
• Any degree-k polynomial in U and U† can be represented as a contraction with matrices in C^(d^k×d^k), enabling its expectation to be computed through the corresponding k-th moment operator.Once the Haar moment operator is known, arbitrary polynomial expectations follow by linear contraction.Source: §I · p. 23
• The moment operator commutes with every U^⊗k, placing it in the commutant of the diagonal representation and motivating its representation-theoretic analysis via Schur–Weyl duality.The section introduces representation-theory tools to characterize this commutant.Source: §I · p. 24
• The Haar k-th moment operator M^(k)_µU(d) is the orthogonal projector onto comm(∆^k[U(d)]), the commutant of the k-order diagonal representation of U(d).This characterization follows from Haar invariance and Hilbert–Schmidt orthogonality, and extends to arbitrary groups equipped with Haar measure.Source: §I · p. 24

1. Representation theory

• Finite-dimensional unitary representations decompose into direct sums of irreducible representations, with equivalent copies forming multiplicity spaces and inequivalent decompositions being multiplicity-free.This decomposition follows by iteratively splitting nontrivial invariant subspaces.Source: §1 · p. 24
• Schur’s lemma states that an operator commuting with an irreducible representation is scalar, while an intertwiner between two irreps is either zero or establishes unitary equivalence up to scale.For Abelian groups, this implies every irrep is one-dimensional.Source: §1 · pp. 24–25

2. Schur-Weyl duality and the commutant of the diagonal action

• The diagonal representation applies the same U ∈ U(d) to each factor of (C^d)⊗k, and it commutes with the symmetric-group representation that permutes tensor components.This commuting structure motivates studying the commutant for Haar-unitary moments.Source: §2 · p. 25
• Schur-Weyl duality states that the diagonal U(d) and symmetric-group representations span each other’s commutants as algebras [53, Theorem 4.2.10].Partitions of k label the irreducible representations, and the joint U(d) × S_k action is multiplicity-free.Source: §2 · p. 26
• The Schur-Weyl decomposition writes (C^d)⊗k as Weyl-module and Specht-module tensor factors, with U(d) acting only on Weyl modules and S_k only on Specht modules.The symmetric and antisymmetric subspaces occur among these components for every k ≥ 2.Source: §2 · p. 26
• Schur-Weyl duality characterizes the unitary k-moment operator through orthogonal projectors onto the spaces W_λ ⊗ S_λ, with coefficients c_λ = Tr(P_λA⊗k)/Tr(P_λ) for symmetric inputs.The derivation uses commutation with the symmetric-group action and Schur’s lemma.Source: §2 · pp. 26–27
• For k = 2, the moment operator lies in the span of identity and swap, equivalently decomposing into symmetric and antisymmetric projectors with trace-determined coefficients.For unit-rank A, only the symmetric-subspace component remains in the general-k decomposition.Source: §2 · pp. 26–27

3. Uniformly random state vectors

• Uniform pure states arise equivalently from the unique U(d)-invariant measure on S(C^d), Haar-random columns U|0⟩, or CP^{d−1} after removing global phase.The complex projective space is identified with pure density matrices modulo U(1).Source: §3 · p. 27
• Moments of degree-k state polynomials reduce to contractions with the moment operator of |ψ⟩⟨ψ|^⊗k, which is proportional to the symmetric-subspace projector.The normalization uses the symmetric-subspace dimension.Source: §3 · p. 27

4. Unitary, spherical and complex-projective k-designs

• A unitary k-design is a distribution whose kth moment operator matches Haar measure; any k-design is also a (k−1)-design for k≥2.Finite subsets qualify when their uniform distribution has this property.Source: §4 · p. 28
• The Clifford group is a unitary 3-design but not a unitary 4-design, so its higher-order commutant exceeds the permutation span [59] [60] [61].The Clifford group is generated by Hadamard, phase, and CNOT gates [58, Theorem 10.6].Source: §4 · p. 28
• Spherical k-designs match the corresponding sphere moments, while their induced projective-state distributions are complex-projective k-designs; unitary k-design orbits provide examples.Applying a unitary k-design to any fixed state produces a complex spherical k-design.Source: §4 · p. 28
• Because Clifford unitaries form a 3-design, stabilizer states are also 3-designs, but not 4-designs [59] [60] [67].The stabilizer-state set is therefore stronger than a 2-design, despite being commonly introduced as one.Source: §4 · p. 28
• Maximal mutually unbiased bases are complex spherical 2-designs; they contain at most d+1 bases, with exactly d+1 in prime-power dimensions, while the general case remains open [70].Two bases are mutually unbiased when every squared overlap equals 1/d.Source: §4 · p. 28
• SIC POVMs consist of d^2 normalized vectors with equal pairwise squared overlap 1/(d+1) and form 2-designs, with explicit constructions known in small dimensions [71, 72].Their symmetry is defined by equality of all pairwise inner products.Source: §4 · p. 29

J. Shadow fidelity estimation

• SFE applies random unitaries from a unitary 3-design, followed by basis measurements, to construct classical-shadow estimators of fidelities to pure target states.The framework reconstructs a least-square state estimator from measurement outcomes; multi-qubit Clifford gates provide an example of suitable unitaries.Source: J. Shadow fidelity estimation · p. 29
• The experiment and post-processing can be separated: outcomes require n˜ρ log n bits, but arbitrary target-state overlaps may be exponentially costly to compute.For stabilizer states and Clifford unitaries, the Gottesman–Knill theorem enables efficient evaluation of the overlap.Source: J. Shadow fidelity estimation · p. 29
• For arbitrary pure target states, SFE uses an unbiased median-of-means estimator that is ε-accurate with confidence 1−δ when n˜ρ ≥160 ε^-2 ln(1/δ) iid state preparations.The median is taken over l = n˜ρ/k means, with k = ⌈8 ln(1/δ)⌉.Source: J. Shadow fidelity estimation · pp. 29, 31
• Compared with DFE, SFE is dimension-independent for arbitrary pure targets but uses randomly selected basis measurements, whereas DFE uses observable expectation values.SFE still needs additional structure, such as stabilizer states, for efficient classical post-processing.Source: J. Shadow fidelity estimation · pp. 29–30
• The dimension-independent guarantee follows because the 3-design structure keeps the estimator variance bounded by Var[ˆfi] < 5 despite individual values scaling as O(d).The proof combines unbiasedness for pure targets, the constant variance bound, and a median-of-means tail inequality.Source: J. Shadow fidelity estimation · pp. 30–31

III. QUANTUM PROCESSES

• Quantum-process certification targets gates, algorithms, and noise processes by characterizing how maps transform quantum states.The tutorial motivates process certification as central to quantum technologies and introduces it after state certification.Source: §III · p. 31
• Quantum processes can be certified through output-state certification or by preparing a process-representing state via maximally entangled inputs.The latter construction is the Choi-Jamiołkowski isomorphism, which operationally represents a process as a quantum state.Source: §III · p. 31
• Direct process-certification methods inherit measurement-feasibility challenges because they require input preparation and output measurement, making SPAM errors a serious obstacle.Self-consistent methods address SPAM robustness but require extensive measurement settings, sampling, and classical post-processing [76] [77] [78].Source: §III · pp. 31–32
• Randomized benchmarking [66] [79] [80] estimates gate performance from error accumulation across random sequences of varying lengths and is robust against SPAM errors.The tutorial presents protocols for gate sets and individual gates, while cross-entropy benchmarking [74] certifies quantum-circuit implementation in quantum-supremacy demonstrations.Source: §III · p. 32

A. Quantum processes and measures of quality

• A quantum channel is a completely positive, trace-preserving map, while complete positivity ensures valid behavior even when the map acts on part of a larger system.The formalism introduces linear maps between operator spaces, including Hermicity preservation, complete positivity, trace preservation, and unitality.Source: A. Quantum processes and measures of… · p. 32

1. The Choi-Jamiołkowski isomorphism

• The Choi-Jamiołkowski isomorphism [82, 83] identifies completely positive maps with bipartite positive semidefinite operators and channels with a constrained subset of states.It provides a practical channel criterion and transfers certification methods from quantum states to processes.Source: §1 · pp. 32–33
• Under the Choi representation, trace preservation corresponds to a maximally mixed input reduction, while complete positivity corresponds to a positive semidefinite Choi matrix.The Choi state can be prepared by applying the channel to a maximally entangled state, and completely positive maps also admit Kraus-operator representations.Source: §1 · p. 33

• The Hilbert-Schmidt inner product of superoperators coincides with the Hilbert-Schmidt inner product of their corresponding Choi matrices.When one map is a unitary channel, the normalized Choi overlap induces entanglement (gate) fidelity.Source: §2 · p. 34
• Average gate fidelity compares two maps by averaging their action on pure input states, while remaining close to 1 when deviations affect only a low-dimensional input subspace.It is defined using the Haar-invariant measure on state vectors and is related to the superoperator inner product [87].Source: §2 · p. 34
• For Hermicity-preserving maps, average gate fidelity is real and symmetric, and for unital completely positive maps average infidelity is nonnegative and vanishes exactly when the channels coincide.Thus, under the stated conditions, infidelity functions as a distance measure between quantum channels.Source: §2 · pp. 34–35
• For trace-preserving and unital channels, average gate fidelity and the Hilbert-Schmidt inner product are affinely related with proportionality scaling O(d^-2), matching entanglement-fidelity scaling.The same identities specialize further when one or both channels are unitary.Source: §2 · p. 35
• The Frobenius norm is an average-case error measure, while the effective depolarizing parameter and χ0,0 process-matrix entry provide additional affinely related quality measures used in randomized benchmarking [88].The effective depolarizing parameter is defined linearly from fidelity for trace-preserving maps and is motivated by the depolarizing channel.Source: §2 · p. 35

• It is a worst-case error measure that supports process certification and error-correction guarantees, but direct diamond-norm certification is typically resource intensive and practically infeasible.The section therefore focuses on relating the diamond norm to average error measures already used in certification.Source: §3 · p. 35
• The diamond norm has strong stability properties, is multiplicative under tensor products, and is sub-multiplicative under products.For Hermitian-preserving maps, the supremum can be attained at a pure density operator; trace-preserving channels are normalized in diamond norm.Source: §3 · p. 36
• The diamond norm captures maximum channel distinguishability, achievable with an ancillary system of the same dimension and a pure input state.For channels X and Y, the distinguishability is 1/2 ∥X − Y∥⋄, optimized over states on an extended input system.Source: §3 · p. 36
• Choi-state trace-norm distinguishability is generally not optimal for distinguishing quantum channels, although the diamond norm and trace norm obey tight bounds.The diamond norm optimizes over ancillary entangled inputs, whereas Choi states correspond to a particular input construction.Source: §3 · p. 36
• For unitary channels, the diamond distance can be computed from the eigenvalues of U†V and the Euclidean distance from their convex hull in the complex plane.This expresses the worst-case optimization through the spectrum of the unitary difference U†V.Source: §3 · pp. 36–37
• The diamond norm can differ substantially from infidelity: general upper bounds contain a dimensional factor O(d) and square-root infidelity scaling, while incoherent channels can achieve linear scaling ∥id − X∥⋄ ∈ O(r(X)) under the stated unitarity condition.For quantum computing, O(d) can scale as O(2^n), and unitary channels show that square-root scaling is unavoidable in general; the linear bound applies to unital X with u(X) = umin + O(r^2(X)).Source: §3 · pp. 37–38

B. Direct quantum process certification

• Choi-state certification lifts direct quantum-state protocols to processes, certifying entanglement gate fidelity through the state fidelity of the process’s Choi state.The process is applied to half of a maximally entangled state, after which the resulting Choi state is certified; Refs. [96] [97] [98] use this strategy [24, 25].Source: B. Direct quantum process certification · p. 38
• Effective measurement strategies can remove the ancillary entanglement requirement by preparing input states, applying the channel, and measuring dichotomic POVMs.These prepare-and-measure schemes implement the effective strategies described in Refs. [96–98].Source: B. Direct quantum process certification · p. 38
• For Clifford operations, the Choi state is a stabilizer state, enabling direct certification with perfect target acceptance and corresponding entanglement-gate-infidelity guarantees [96, Proposition 3].The protocol applies stabilizer-state certification to J(C) and can be implemented as a prepare-and-measure scheme.Source: B. Direct quantum process certification · pp. 38–39

Further reading

• Further work extends direct certification to non-trace-preserving processes, measurements, multi-qudit systems, projective 2-designs, and entanglement-property detection [96] [97] [98].Related fidelity-estimation approaches sample process outputs from Pauli eigenstates or complex projective 2-designs and study required input-state counts [43] [50] [102].Source: Further reading · p. 39

C. Randomized benchmarking

• Randomized benchmarking reduces sensitivity to SPAM errors by applying processes in sequences of varying lengths, where process effects change with length while preparation and measurement errors enter linearly and independently of length.Fitting signals across sequence lengths reveals process properties disentangled from SPAM errors [66] [79] [80] [103] [104].Source: C. Randomized benchmarking · p. 39
• Prototypical randomized benchmarking samples group-element sequences, appends an inverse, measures the output, and fits decay parameters from data collected over multiple sequence lengths.The resulting RB parameters are extracted from the empirical sequence-length dependence.Source: Figure 4 · p. 40, C. Randomized benchmarking · p. 40
• For a unitary 2-design with gate-independent noise, twirling maps trace-preserving channels to depolarizing channels, yielding a single-exponential RB model whose decay parameter estimates effective depolarization and average gate fidelity.SPAM errors affect only fitting constants A and B, not the resulting fidelity estimate.Source: C. Randomized benchmarking · p. 41
• Rigorous RB guarantees require accounting for measurement, sequence-subsampling, and exponential-fitting errors; practical scalable bounds depend on sequence variance and efficient group processing.Clifford-group analyses provide favorable sampling complexities, while efficient inversion follows from Gottesman–Knill and the group’s design property [93, 106–111].Source: C. Randomized benchmarking · pp. 41–42

Protocol

• Interleaved RB [112] interleaves a specific target gate with uniformly random group gates, computes the modified-sequence inverse, and outputs decay parameters from both experiments.The target sequence is g = (g1, gT, g2, gT, g3, . . . , gT, gm), with gi ∈ G drawn uniformly at random.Source: Interleaved randomized benchmarking · p. 42

Analysis

• Under mostly gate-independent noise and a unitary 2-design, the modified experiment follows a single-exponential model estimating the effective depolarizing parameter of the target gate concatenated with the common error channel.The standard experiment estimates p(Λ), while the interleaved experiment estimates p(G†T φ(gT)Λ) and the corresponding error-map fidelity.Source: Interleaved randomized benchmarking · pp. 42–43

Error bounds

• The composite-channel bound has small systematic error when u(Λ) ≈ p(Λ)^2, which holds for decoherent noise, enabling estimation of the target gate’s average gate fidelity.The bound uses an estimate of the unitarity ˆu(Λ) and controls the systematic error through the right-hand side of Eq. (262).Source: Interleaved randomized benchmarking · p. 43

Certification

• Interleaved RB can certify individual-gate average performance, but estimating the target gate requires controlling the common noise channel’s unitarity or assuming sufficiently incoherent errors.Unitarity variants of RB provide estimates, while the simpler estimator has systematic error controlled by ϵ under the incoherence condition.Source: Interleaved randomized benchmarking · p. 43

Worst-case bounds

• Estimating unitarity also yields relevant worst-case performance bounds from average gate fidelities using Theorem 48.This extends the role of unitarity beyond tighter interleaved-RB error bounds.Source: Interleaved randomized benchmarking · p. 43

Further reading

• Randomized benchmarking originated for unitary groups and Clifford gates [66] [79] [80] [103] [104], but flexible protocols extend to groups that are not unitary 2-designs.General-group protocols support T-gates, leakage characterization, and multiple decay components from irreducible representations [120] [121] [122] [123] [124] [125] [126] [127].Source: Further reading · pp. 43–44
• RB developments address gate-dependent noise, multiple decays, individual-gate certification, and alternative sequence distributions, with rigorous general fitting frameworks based on representation-theoretic conditions [75, 116].Character benchmarking projects data onto irreducible representations, while interleaved RB provides stronger individual-gate certificates when errors are sufficiently incoherent [88] [129].Source: Further reading · pp. 43–44

D. Cross-entropy benchmarking

• XEB has variants for average gate fidelities and for certifying sampling by one specific circuit; the latter is an application-layer protocol close to the physical layer [144].Source: D. Cross-entropy benchmarking · p. 44
• Cross-entropy benchmarking (XEB) [74] certifies whether a quantum circuit samples from its intended output distribution, supporting demonstrations of quantum supremacy [144].It targets circuits whose outcome probabilities remain classically calculable for validation.Source: D. Cross-entropy benchmarking · pp. 44–45

Cross-entropy and cross-entropy fidelity

• For random-circuit sampling, Porter–Thomas-shaped distributions motivate cross-entropy benchmarks, because Haar-random unitary measurement probabilities approach this distribution at large dimension [149].The tutorial gives H(pU) = ln(d) + γ −1 and HX(puni, pU) = ln(d) + γ for Porter–Thomas pU.Source: Cross-entropy and cross-entropy fidelity · p. 45
• The cross-entropy difference compares a distribution’s excess cross-entropy beyond the uniform distribution, satisfies dXE(puni, p) = 0, and equals 1 for Porter–Thomas p.Other distributions can score higher than p itself, limiting its interpretation.Source: Cross-entropy and cross-entropy fidelity · pp. 45–46
• Cross-entropy fidelity is a linear proxy for cross-entropy difference, normalized so FX(puni, p) = 0 and expected to reach FX(pU, pU) = 1 for Porter–Thomas circuits.For unitary 2-designs, the expectation reproduces the Porter–Thomas value up to 1 + O(1/d).Source: Cross-entropy and cross-entropy fidelity · p. 46

Cross-entropy benchmarking protocol

• Cross-entropy benchmarking [74] [144] samples a circuit repeatedly, computes each observed outcome’s ideal probability classically, and applies fF(p)=dp−1 or fH(p)=−ln(p).The protocol estimates cross-entropy fidelity or cross-entropy from measurement outcomes and ideal probabilities.Source: Cross-entropy benchmarking protocol · p. 46
• XEB cannot directly handle non-classically simulable circuits because step 2 requires individual ideal probabilities; restricted tractable circuits can instead support extrapolation.This limitation motivates evaluating subclasses that remain simulable on powerful classical computers.Source: Cross-entropy benchmarking protocol · p. 46
• For Haar-random targets, linear XEB admits sampling guarantees based on bounded ideal probabilities, Porter–Thomas behavior, moment bounds, union bounds, and Hoeffding’s inequality [151].The ideal probabilities are bounded with high probability, enabling concentration of the estimator.Source: Cross-entropy benchmarking protocol · pp. 46–47
• Estimating cross-entropy requires additional lower-tail control because the logarithmic score is not bounded by an upper bound on ideal probabilities alone [151].The analysis therefore uses the explicit tail distribution rather than only moment bounds.Source: Cross-entropy benchmarking protocol · p. 47
• Without assumptions, cross-entropy does not imply total-variation closeness; if noise increases entropy, Pinsker’s inequality yields a bound through DXE(q,p)=HX(q,p)−H(p).DXE can be estimated with XEB by obtaining the entropy shift analytically or numerically.Source: Cross-entropy benchmarking protocol · p. 48

Further reading

• Non-interactive black-box verification is difficult because distributions that are classically hard to sample are also difficult to verify efficiently from samples alone [169] [170] [171].An extensive overview of verification and certification methods is provided in Ref. [151].Source: Further reading · p. 48
• Quantum-supremacy verification includes HOG [173], BOG [146], trap-based circuits [174], and state- or process-certification approaches, but these methods generally retain simulation or assumption requirements.Classical spoofing strategies and evidence for spoofing hardness remain active research topics [175] [176] [177].Source: Further reading · p. 48

ACRONYMS

• PVM denotes a projector-valued measure.Source: ACRONYMS · p. 49