Efficient verification of quantum processes

• O(4^n) measurements make standard quantum process tomography impractical for intermediate-size n-qubit devices.Source: §I · p. 1
• Existing alternatives, including compressed sensing, direct fidelity estimation, randomized benchmarking, and accreditation, retain resource or applicability limitations.Source: §I · p. 1
• Quantum state verification provides a model based on efficient local pass-or-fail tests for confirming target states.Source: §I · p. 1
• Quantum process verification is proposed through two protocols derived by relating processes to states via the Choi-Jamiołkowski isomorphism.The protocols are ancilla-assisted and prepare-and-measure, with the latter requiring no ancilla systems.Source: §I · p. 1
• The protocols demonstrate exponential improvement over QPT, quadratic improvement over DFE, local-measurement implementation, and adaptation to quantum-measurement verification.Source: §I · p. 2

• Applying a process to one half of an entangled state produces its Choi matrix, from which the process can be recovered.Source: §II · p. 2
• The Choi-Jamiołkowski isomorphism maps Choi matrices to linear maps and identifies complete positivity with positive semidefiniteness.Source: §II · p. 2
• The framework does not require trace preservation, allowing treatment of postselection and particle losses.Source: §II · p. 2
• Determining the Choi matrix determines all information about the process, motivating ancilla-assisted quantum process tomography with fixed entangled inputs.Source: §II · p. 2

• A unitary gate can be verified by applying QSV to its corresponding pure maximally entangled Choi state.Source: §III · p. 2
• The gate-verification task distinguishes entanglement fidelity 1 from fidelity at most 1−ϵ using confidence 1−δ, with sample complexity as the resource measure.Source: §III · p. 2
• For any quantum gate U, verification to entanglement fidelity 1−ϵ and confidence 1−δ requires N ≈ 1/ν(Ω) ϵ^-1 ln δ^-1 input states.Source: §III · p. 2
• Trace-preserving constraints can improve efficiency, while relaxing trace preservation makes the relevant bound attainable.Source: §III · p. 2
• The indirect Choi-state strategy is named ancilla-assisted quantum process verification and extends to processes with pure corresponding Choi states.Source: §III · p. 2

• A one-way adaptive ancilla-assisted protocol can be converted to a prepare-and-measure protocol without efficiency loss.Source: §IV · p. 3
• The prepare-and-measure protocol randomly selects an input state and applies a corresponding pass-or-fail measurement to the output.Source: §IV · p. 3
• The conversion preserves the ancilla-assisted worst-case failure bound, yielding equal efficiency for verifying a quantum gate.Source: §IV · p. 3
• Ancilla-assisted verification uses one entangled input kind, whereas prepare-and-measure verification uses no ancilla but requires multiple input-state kinds.Source: §IV · p. 3
• General spherical 2-designs can provide experimentally convenient input states in unrestricted settings.Source: §IV · p. 3

Applications

• The protocols verify cnot, arbitrary Clifford circuits, n-qubit C^(n−1)Z and C^(n−1)X gates, including Toffoli gates, as well as processes from the Deutsch-Jozsa algorithm.The C^(n−1)X verification is equivalent to that of C^(n−1)Z up to local unitary transformations.Source: §V · pp. 3–4

cnot and Clifford circuits

• The cnot verification can be improved by employing more measurement settings, and Clifford circuits become universal for quantum computation with suitable state preparations.Source: §V · p. 4

Clifford circuits

• For any n-qubit Clifford circuit, the protocols provide verification to entanglement fidelity 1−ϵ and confidence 1−δ using only local Pauli measurements.The Choi state is a stabilizer state, enabling systematic verification through its stabilizers.Source: §V · p. 4
• Quadratic speedup over DFE is obtained for Clifford-circuit verification, whose DFE sample complexity is O(ϵ−2 log δ−1).The protocols are also quadratically more efficient than randomized benchmarking when sample complexity is considered.Source: §V · p. 4

Controlled gates

• N ≈(n + 1)ϵ−1 ln δ−1 input states verify n-qubit C^(n−1)Z gates to fidelity 1−ϵ and confidence 1−δ.The output Choi state is a hypergraph state with an (n + 1)-coloring, giving spectral gap ν(Ω)=1/(n + 1).Source: §V · p. 4, Figure 1 · p. 4

Physical setting

• When processes are not trace preserving because of postselection or particle losses, verification is conditioned on obtaining an output state.The postselected Choi state is not guaranteed to be produced for every input.Source: §VI · pp. 4–5

Scope and assumptions

• Under postselection, the process can be verified only up to a constant factor c because ˜K is invariant under K→cK.The number of postselected output states is therefore used as the resource measure.Source: §VI · p. 5
• Relaxing trace preservation allows the corresponding Choi state to be any quantum state, while trace-preserving processes impose an additional reduced-state constraint.Source: §VI · p. 5

Postselected processes

• N ≈ν(Ω)^−1ϵ−1 ln δ−1 postselected output states suffice to verify a postselected process to entanglement fidelity 1−ϵ and confidence 1−δ.This follows by applying QSV to the process’s postselected Choi state.Source: §VI · p. 5

Protocol conversion

• The AAPV protocol converts to a PMPV protocol with equal efficiency for processes with postselection.The conversion applies to one-way adaptive QSV protocols.Source: §VI · pp. 5–6

Measurement verification

• Measurement verification differs from process verification because reliable quantum measurements cannot be assumed available for verifying the measurement device itself.Source: §VII · p. 6

Projective measurements

• N ≈ϵ−1 ln δ−1 input states suffice to verify the projective measurement P.The protocol randomly prepares |i⟩ with probability 1/d and passes when measurement M returns outcome i.Source: §VII · p. 6

Entangled measurements

• For entangled measurements, verification can use entangled inputs or treat the measurement as a postselected quantum process when single-partite measurements are trusted.Source: §VII · p. 6

Summary

• Quantum process verification provides an exponential improvement over quantum process tomography and a quadratic improvement over direct fidelity estimation.Source: §VIII · p. 6
• The protocols verify quantum gates, quantum circuits, and processes of quantum algorithms, and can also efficiently verify quantum measurements.Source: §VIII · p. 6
• Only local measurements are required, making the protocols compatible with current experimental techniques.The paper identifies state-preparation and measurement errors as an important direction for future modification.Source: §VIII · p. 6

• The appendix presents verification of more quantum gates using the AAPV and PMPV protocols.Source: Appendix A · p. 6

1. Identity

• The identity process is trivial, with Choi matrix equal to the maximally entangled state itself.Source: §1 · p. 6
• The identity AAPV protocol has spectral gap ν(ΩΥI) = 1/2, and its PMPV verification operator specifies prepare-and-measure operators with their probability distributions.Source: §1 · p. 7
• The identity protocol ΩΥI is not optimal; using more measurement settings improves its efficiency to ν(ΩΥI) = 2/3.Source: §1 · p. 7

2. The bit-flip gate

• The same protocols extend directly to any single-qubit gate U through the stated substitutions, including the bit-flip, Hadamard, and phase gates.Source: §1 · p. 7, §2 · p. 7, §3 · p. 7

3. The Hadamard gate

• The bit-flip, Hadamard, and phase-gate AAPV protocols each have spectral gap 1/2, with corresponding PMPV protocols.Source: §2 · p. 7, §3 · p. 7, §4 · p. 7

• The Deutsch-Jozsa algorithm is verified for representative constant and balanced functions using the proposed protocols.Source: Appendix B · p. 7
• For f(x) = 0, the process is I⊗(n+1), while for f(x) = 1 it is I⊗n ⊗Z and can be efficiently verified.Source: Appendix B · p. 7
• The constant-function AAPV construction uses six stabilizer generators and has spectral gap ν(ΩΥf(x)=1) = 1/6.Source: Appendix B · pp. 7–8
• N ≈6ϵ−1 ln δ−1 input states are required for fidelity 1−ϵ and confidence 1−δ in that representative constant case.Source: Appendix B · p. 8
• For a representative balanced case, the three-qubit process I ⊗CNOT is determined by the second qubit, with f(x) = x2.Source: Appendix B · p. 8