Efficient evaluation of quantum observables using entangled measurements

VQE context

• VQE minimizes a Hamiltonian expectation value by varying parameters of a quantum state and estimating weighted sums of Pauli-string expectation values.Qubit Hamiltonians are linear combinations of tensor products of Pauli operators, and quantum circuits evaluate the individual Pauli-string expectations.Source: INTRODUCTION · p. 1

Motivation

• Quantum-classical hybrid algorithms require many circuit executions to estimate observables, with measurement counts described as astronomically large for quantum chemistry applications.VQE uses nested outer, middle, and inner iterations, and the inner iteration requires O(ϵ^-2) samples for statistical error ϵ.Source: INTRODUCTION · pp. 1–2

Existing grouping

• TPB grouping uses joint measurability of commuting Pauli strings, while graph-coloring heuristics such as LDFC construct practical groups despite the underlying NP-complete coloring problem.The Pauli graph represents TPB incompatibility, and LDFC is used to obtain groups of jointly measurable strings.Source: INTRODUCTION · p. 2

Contributions

• Entangled measurements supplement TPB grouping to reduce the number of jointly measurable Pauli-string groups while accounting for covariance and added CNOT-gate errors.The paper evaluates the approach theoretically and experimentally and proposes sampling based on group sizes to mitigate covariance effects.Source: INTRODUCTION · p. 2

Experimental scope

• The paper demonstrates its approach through a proof-of-concept experiment on a simple Hamiltonian using real quantum computers.The stated contributions include evaluating errors from additional CNOT gates and experimentally demonstrating the method.Source: INTRODUCTION · p. 2

Evaluation of Pauli strings

• The chosen fermion-to-qubit transformation can affect grouping results because Jordan–Wigner, parity, and Bravyi–Kitaev mappings produce different Pauli-string structures.The paper presents the fermionic Hamiltonian and identifies these three transformations as standard mappings to qubit systems.Source: Evaluation of Pauli strings · p. 2
• Pauli-string expectation values are obtained by measuring in the computational basis after single-qubit rotations implement other measurement bases.A target observable is a linear combination of N-qubit Pauli strings, with fermionic systems mapped to qubits by transformations such as Jordan–Wigner, parity, or Bravyi–Kitaev.Source: Evaluation of Pauli strings · p. 2

Grouping Pauli strings with TPB

• TPB grouping collects Pauli strings simultaneously diagonalizable in tensor-product X, Y, or Z bases, using Pauli-graph coloring and LDFC to construct groups.Graph nodes represent Pauli strings, edges connect strings incompatible under TPB, and colors identify jointly measurable groups.Source: Grouping Pauli strings with TPB · pp. 2–3

Grouping with TPB and entangled measurements

• Bell measurements jointly measure σxσx, σyσy, and σzσz, which require three TPB measurement types when measured separately.Simultaneous diagonalization supports joint measurements using entangled observables, but larger entanglements increase circuit-construction cost and depth.Source: Grouping with TPB and entangled… · p. 3
• The proposed two-phase method first selects entangled measurements, then groups Pauli strings using TPB together with those measurements.Preconstructed circuits reduce circuit-construction cost in the grouping phase, while merge history constrains later measurement choices.Source: Grouping with TPB and entangled… · p. 3

Standard error caused by the grouping of Pauli strings

• When samples per group scale with group size, grouping has smaller standard error than no-grouping at equal total sample counts despite covariance effects.The paper also notes that additional covariance from grouping may require more measurements in some settings.Source: Standard error caused by the grouping… · p. 3

Number of groups

• Entangled-measurement groupings achieved fewer groups than Pauli-graph clique sizes, although TPB+2Q and ALL did not consistently outperform TPB+Bell.For Jordan–Wigner transformations, TPB+Bell produced fewer groups for most molecules than TPB and TPB+2Q; the comparison covers LiH, BeH2, H2O, NH3, and HCl.Source: Number of groups · pp. 3–4, Table I · p. 4
• Entangled measurements reduce measurement counts, but two-qubit gate errors and added CNOT gates constrain their practicality on NISQ computers.The method keeps circuit depth constant for fixed entangled measurements and allows added depth to be adjusted for multi-qubit-gate errors.Source: Grouping with TPB and entangled… · p. 3, Number of groups · p. 4

Effect of additional CNOT gates

• Comparable expectation values with less standard error were observed up to two-qubit error 0.010, while larger errors made Bell grouping worse because of its extra CNOT gate.The comparison used three no-grouping groups versus one Bell-measurement group.Source: Effect of additional CNOT gates · pp. 4–5
• On IBM Q Tokyo and IBM Q Poughkeepsie, Bell grouping again achieved comparable expectation values with less standard error than no-grouping.The experiments suggest readout error had a greater influence than the additional two-qubit error.Source: Effect of additional CNOT gates · p. 5, Table III · p. 6

VQE with entangled measurements

• VQE with a Bell measurement used one circuit per iteration instead of three and required fewer circuits to converge than no-grouping.The experiment used 40 iteration steps and fixed 8192 executions per circuit job.Source: VQE with entangled measurements · p. 5, Figure 1 · p. 6

DISCUSSION

• TPB+Bell matched the number of groups from all-measurement grouping for some Jordan–Wigner molecules while keeping additional CNOT depth constant at 1.Jordan–Wigner Hamiltonians contain an even number of σx and σy operators, a structure suited to Bell measurements.Source: DISCUSSION · p. 5
• The method can address nearest-neighbor hardware by adapting grouping or qubit layout; a one-dimensional lattice is sufficient for the measurement circuits.The two-qubit measurement strategy needs one nonlocal gate per two qubits under the layout approach.Source: DISCUSSION · p. 5
• Reducing groups can improve cloud-job execution time because execution time is roughly proportional to the total number of circuits.The proposal combines TPB with only part of the available entangled measurements.Source: DISCUSSION · p. 5

Entangled measurements

• Entangled measurements measure entangled observables, which are described by nonseparable positive operator-valued measures.Observable entanglement can be detected through violation of the dual Bell-CHSH inequality.Source: Entangled measurements · p. 7
• Bell measurements are projective measurements on Bell states and are used in quantum teleportation.Bell states are maximally entangled states.Source: Entangled measurements · p. 7
• A Bell measurement yields expectation values of σx ⊗ σx, σy ⊗ σy, and σz ⊗ σz from the same measurement result.These Pauli products are jointly measurable with Bell measurements even though TPB cannot obtain them simultaneously.Source: Entangled measurements · p. 7
• The Bell-measurement circuit uses CNOT and Hadamard gates and is the inverse process of creating Bell states.The circuit is shown in Fig. 2.Source: Entangled measurements · p. 7, Figure 2 · p. 7

Grouping algorithm

• The algorithm constructs a TPB Pauli graph, orders strings by descending degree, and greedily merges jointly measurable nodes into observable groups.Merged nodes are returned as the final groups.Source: Grouping algorithm · p. 7
• Compatibility is tested by checking whether available entangled measurements fit uncovered qubit positions under permutations.The assignment procedure updates measurement coverage as compatible positions are added.Source: Grouping algorithm · pp. 7–8
• TPB+Bell uses Bell and TPB measurements, whereas TPB+2Q additionally includes Omega-XX, Omega-YY, Omega-ZZ, and Chi measurements.The resulting group count depends on the order in which measurements are applied.Source: Grouping algorithm · p. 7

COMPETING INTERESTS

• The work includes a patent filing with the Japan Patent Office and a planned filing with the US Patent and Trademark Office.Source: COMPETING INTERESTS · p. 8

Supplementary Information for “Efficient evaluation of quantum

• The supplementary information accompanies the paper on efficient observable evaluation using entangled measurements.Source: Supplementary Information for… · p. 10

Appendix A: Pauli graph and LDFC algorithm

• The Pauli graph connects Pauli-string nodes exactly when they are not qubit-wise commutative.Its construction checks every pair of strings for any noncommuting single-qubit operators.Source: Appendix A · p. 10
• Largest degree first coloring sorts nodes by descending degree and assigns each node the smallest color absent from its neighbors.Source: Appendix A · p. 10

Appendix B: Examples of entangled measurements

• Omega-YY measurements jointly measure σy ⊗σy, σx ⊗σz, and σz ⊗σx.The corresponding circuit is shown in Fig. S1.Source: Appendix B · pp. 10–11, Figure S1 · p. 11
• Omega-ZZ measurements jointly measure σz ⊗σz, σx ⊗σx, and σy ⊗σx, whereas omega-XX measurements jointly measure σx ⊗σx, σy ⊗σz, and σz ⊗σy.Their circuit implementations are shown in Fig. S2.Source: Appendix B · pp. 11–12, Figure S2 · p. 12
• Chi measurements jointly measure σx ⊗σy, σy ⊗σz, and σz ⊗σx.The chi-measurement circuit is shown in Fig. S3.Source: Appendix B · p. 12, Figure S3 · p. 12
• The experiments use only two-qubit entangled measurements because multipartite circuits generally require greater depth and may incur more errors on current quantum computers.Source: Appendix B · p. 12

• The grouped standard-error analysis includes covariance terms, whereas the no-grouping expression does not contain covariance effects.Source: §1 · p. 13
• The proposed allocation sets each group’s samples proportional to its size, S_k = |s_k|S, while keeping total samples equal between grouping and no-grouping.Source: §1 · p. 13
• Theorem 1 states that the standard error with grouping is less than or equal to the standard error without grouping.Source: §1 · p. 13

• Bell-measurement variance includes covariance terms between the three Heisenberg Pauli observables, unlike independent measurement variance.Source: §2 · p. 14
• 0.00159 versus 0.00479 average squared standard error: Bell-measurement grouping outperformed no-grouping across 10,000 Haar-random states.The experiment used 1,500 samples per Bell-measurement group versus 500 samples per Pauli string for three ungrouped groups.Source: §2 · p. 14
• Figure S4 compares the distributions of squared standard errors for the two measurement strategies over the random states.Source: Figure S4 · p. 14

• When both methods converged, the proposed method converged faster than the baseline, although some experiments failed to converge.The experiments ran on IBM quantum computers, and outcomes depended on device conditions during queue delays.Source: Appendix D · p. 14, Figure S5 · p. 15
• SPSA was not robust enough to noise because some energies decreased and then increased, so converged values were not always minima.The authors suggest testing other optimizers and using checkpoints to recover from detected fluctuations.Source: Appendix D · p. 15