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Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers
William J. Huggins, Jarrod McClean, Nicholas Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, Ryan Babbush
TL;DR
VQE measurement costs can make larger-molecule applications appear infeasible because accurate Hamiltonian estimates require many circuit repetitions. The paper uses low-rank factorization of the two-electron integral tensor to group measurements, then evaluates measurement reductions and error mitigation in noisy simulations. For the largest systems studied, the approach reduces measurement requirements by more than three orders of magnitude relative to commonly used bounds, while its deeper circuits are largely balanced by reduced readout errors and postselection.
Problem
Prior measurement bounds suggested that applying variational algorithms to larger molecular systems might require infeasibly many circuit repetitions.
Method
The method factorizes the two-electron integral tensor to create measurement groupings and uses basis rotations that enable direct postselection on particle number and spin.
Results
More than three orders of magnitude speedup over commonly used measurement bounds was observed for the largest Hydrogen, water, and Nitrogen systems studied.
Takeaways & Limitations
The strategy combines substantially reduced measurement time with efficient error mitigation for variational quantum simulations of molecular systems.
Takeaways & Limitations
For systems without truncatable tensor factorization, the scheme may require N^2 + 1 partitions, and estimating the full fermionic 2-particle reduced density matrix requires O(N^2) measurement bases.
Abstract
from arXiv · showhide
Variational algorithms are a promising paradigm for utilizing near-term quantum devices for modeling electronic states of molecular systems. However, previous bounds on the measurement time required have suggested that the application of these techniques to larger molecules might be infeasible. We present a measurement strategy based on a low rank factorization of the two-electron integral tensor. Our approach provides a cubic reduction in term groupings over prior state-of-the-art and enables measurement times three orders of magnitude smaller than those suggested by commonly referenced bounds for the largest systems we consider. Although our technique requires execution of a linear-depth circuit prior to measurement, this is compensated for by eliminating challenges associated with sampling non-local Jordan-Wigner transformed operators in the presence of measurement error, while enabling a powerful form of error mitigation based on efficient postselection. We numerically characterize these benefits with noisy quantum circuit simulations for ground state energies of strongly correlated electronic systems.
I. INTRODUCTION
VQE uses a quantum device to prepare and measure parameterized molecular wavefunctions, but accurate Hamiltonian averaging can require prohibitively many circuit repetitions. The paper introduces measurement and error-mitigation strategies that reduce grouping and sampling burdens while avoiding non-local Jordan–Wigner measurement challenges.
- VQE combines parameterized quantum-state preparation and observable measurements with classical optimization to approximate molecular ground-state wavefunctions and energies.
- Hamiltonian averaging decomposes the Hamiltonian into Pauli strings whose expectation values are estimated independently by repeated measurement.
- Prior measurement bounds led to claims that chemistry applications require “a number of measurements which is astronomically large.”
- The proposed approach uses two-electron integral tensor decomposition rather than Pauli-string properties and evaluates grouping performance through simulated estimator variances.
- Cubic improvement relative to recent strategies and up to three orders of magnitude fewer total measurements arise when grouping covariances reduce overall variance.
- The method retains Jordan–Wigner encoding while estimating 1- and 2-particle fermionic expectations through only 1-local and 2-local qubit measurements.
- Direct postselection on total particle number η and z-component of spin Sz provides an additional error-mitigation opportunity.
A. Using Hamiltonian Factorization for Measurements
The measurement strategy factorizes the electronic-structure Hamiltonian and rotates the quantum state into single-particle bases before computational-basis measurement. This enables simultaneous sampling of diagonal number operators, with basis changes implemented by linear-depth circuits.
- Tensor factorization techniques are applied directly to the Hamiltonian measurement problem.
- Eigendecomposition of the two-electron integral tensor provides a controllable low-rank approximation by discarding small eigenvalues.
- Basis Rotation Grouping applies each single-particle basis-change circuit before measurement and simultaneously samples all ⟨np⟩ and ⟨npnq⟩ values in that basis.
- Because np = (11 + Zp)/2 is diagonal under Jordan–Wigner, computational-basis measurements access all required diagonal qubit operators simultaneously.
- Each basis change uses N^2/4−N/2 two-qubit gates and exactly N depth on a linear qubit array, assuming a total-spin eigenstate.
- The authors quantify measurement time by circuit repetitions because hardware platforms have different repetition rates.
B. Circuit Repetitions Required for Energy Measurement
The Basis Rotation Grouping strategy reduces circuit repetitions for energy estimation by grouping terms through basis rotations, outperforming alternatives across many molecular systems. Its largest advantages appear for growing hydrogen chains, while CISD-based measurement allocation adds less than 3% measurement time.
- Measurement strategies: Basis Rotation Grouping compares against separate measurements, Pauli Word Grouping, RDM-constrained grouping, and a Hamiltonian-coefficient bound.Figure 1 evaluates these strategies on hydrogen chains, water, and nitrogen systems at a 2σ precision of 1.0 millihartree.
- Scaling and variance: For hydrogen chains with more than four fermionic modes, Basis Rotation Grouping consistently requires significantly fewer measurements than the other strategies.Its apparent asymptotic scaling advantage becomes more pronounced with increasing hydrogen-chain length and basis-set size.
- Scaling and variance: The method provides no measurement-time benefit over heuristic grouping for the minimal-basis water molecule.The reported advantage is therefore system-dependent rather than uniform across all molecular cases considered.
- Measurement allocation: Using CISD variances to allocate repetitions increases measurement time by less than 3% compared with allocation based on the true ground state.The comparison uses ratios of approximate to optimal allocation times for the systems and measurement techniques studied.
C. Error Mitigation
The measurement strategy reduces readout-error sensitivity by using one- and two-qubit operators and enables postselection onto desired particle-number and spin sectors. Simulations compare these benefits with Pauli-word grouping and show a trade-off from the added basis-transformation circuit, especially when state-preparation noise is included.
- Readout-error mitigation: Measuring one- and two-qubit operators replaces O(N)-qubit Jordan–Wigner measurements, reducing the exponential suppression of Pauli-string estimates under bitflip noise.The bitflip channel biases a K-qubit Pauli-string estimator toward zero by a factor exponential in K.
- Symmetry postselection: Each prescribed measurement also measures total particle number η and spin z component Sz, enabling postselection on desired quantum numbers.Postselection discards outcomes outside the desired subspace and evaluates observables using the remaining records.
- Symmetry postselection: Postselection costs approximately 1/Tr(Pρ) additional measurements, although discarding incorrect particle-number sectors may reduce the observed variance.The variance can decrease because measurements from energetically distinct sectors are excluded.
- Comparison with prior mitigation: Earlier post-processing approaches target particle-number parity but do not easily project onto the correct η and Sz eigenvalues and can require more measurements than this strategy.Their Hamiltonian-times-parity products also introduce additional non-simultaneously measurable terms.
- Noisy-circuit comparison: Compared with Pauli Word Grouping, Basis Rotation Grouping uses 30 additional two-qubit gates but has comparable mitigated errors in some regimes and lower errors when measurement noise dominates.The comparison is reported for stretched six-hydrogen-chain simulations under the specified circuit and readout noise models.
- Noisy-circuit comparison: With state-preparation noise included, absolute errors remain larger than the usual chemical-accuracy target of approximately 1 mHa even at the lowest considered noise levels.The authors state that practical VQE implementations will require combining multiple error-mitigation methods.
III. DISCUSSION
The paper presents a low-rank measurement strategy that substantially reduces repetitions for quantum-chemistry energies while improving resilience to measurement errors. Its benefits are demonstrated across molecular systems, with scope extending to broader fermionic simulations but with increased partition costs for some tasks.
- III. DISCUSSION: O(N) distinct measurement sets replace the O(N^4) sets implied by naive term counting.The strategy rewrites the Hamiltonian using factorizations of the two-electron integral tensor.
- III. DISCUSSION: More than 10-fold speedups occur for the largest Hydrogen-chain, water, and Nitrogen systems versus recent state-of-the-art strategies.The largest systems include a 24-qubit Hydrogen chain and water simulations and 20-qubit Nitrogen calculations.
- III. DISCUSSION: More than 10^3-fold speedups occur relative to commonly used measurement bounds in the literature.For a 24-qubit, six-Hydrogen chain example, the bound corresponds to about 55 days, whereas the proposed approach requires 44 minutes at 10 kHz.
- III. DISCUSSION: Noisy-circuit simulations show that reduced readout errors and efficient error mitigation largely offset the deeper circuits required by basis rotations.The comparison includes quantum subspace expansion as a moderately expensive mitigation strategy.
- III. DISCUSSION: Without truncating the tensor factorization, the scheme requires N^2 + 1 partitions, while estimating the full fermionic 2-particle reduced density matrix requires O(N^2) bases.These costs apply to general fermionic systems and to reduced-density-matrix estimation rather than energy estimation alone.
Appendix A: Variance Bounds
Appendix A shows that measurement bounds depend strongly on whether the Hamiltonian is represented with fermionic operators or Jordan-Wigner-transformed qubit operators. Cancellations and lower variances in the qubit representation make its bounds considerably tighter, while variance approximations can track actual variance closely.
- Variance-bound construction: The standard measurement bound estimates repetitions from Hamiltonian coefficients after decomposing the Hamiltonian into independently measured Pauli strings.The bound assumes each Pauli measurement has variance at most one.
- Fermionic representation: Fermionic coefficient bounds can be obtained after normal ordering, but counting terms and Hermitian conjugates requires care.The resulting fermionic bound is denoted M_f and is described as looser than necessary in multiple ways.
- Representation effects: Number-operator terms have half the coefficient magnitude in the qubit representation because Jordan-Wigner mapping introduces constant contributions that do not affect variance.Number operators have eigenvalues 0 and 1, giving lower maximum variance than Pauli operators with eigenvalues -1 and 1.
- Representation effects: Two-body terms exhibit cancellations, including an exact factor-of-two reduction for class-V coefficient magnitudes in the qubit representation.Class-V cancellation follows from an eight-fold symmetry, while class-IV differences arise from analogous cancellations without an obvious symmetry.
- Representation effects: Combining all five classes makes the qubit coefficient magnitude roughly half the sum of individual partition magnitudes.Opposite-sign Z and ZZ contributions from one- and two-number-operator terms explain an important part of this cancellation.
- Numerical comparison: Qubit-derived variance bounds are considerably tighter than fermionic bounds because of these cancellation effects.The comparison is reported for an eight-hydrogen chain and includes an improved fermionic bound.
- Numerical comparison: FVA and QVA are variance approximations rather than guaranteed upper bounds, computed from fermionic and qubit Hamiltonian representations respectively.QVA values are reported as nearly identical to the actual variance expected when measuring Jordan-Wigner-transformed terms separately.
Appendix B: Applying the Fermionic RDM Constraints to the Qubit Hamiltonian
Appendix B tests reduced-density-matrix constraints on fermionic and qubit Hamiltonian representations using observed ground-state variance. Although representation-dependent bounds differ substantially, applying the constraints to the qubit Hamiltonian has at most a marginal effect on the measured variance.
- RDM-constrained Hamiltonians: RDM constraints construct a modified Hamiltonian with the same expectation value but a lower bounded maximum variance.The minimization is performed using known n-representability constraints and standard linear programming.
- Evaluation: The comparison uses actual ground-state variance rather than variance bounds or approximations.The same Pauli Word Grouping strategy is applied to the fermionic and qubit constrained Hamiltonians.
- Evaluation: Applying RDM constraints to the qubit Hamiltonian instead of the fermionic Hamiltonian changes the observed variance marginally at best.This result contrasts with the substantial difference between variance bounds formulated in the two representations.
Appendix C: Low Rank Decomposition
Appendix C derives the low-rank decomposition by reshaping the two-electron integral tensor into a matrix, eigendecomposing it, and diagonalizing the resulting one-body operators in rotated orbital bases. Real spatial orbitals provide the symmetry used in this construction.
- Hamiltonian preparation: The decomposition starts from the chemist’s standard form of the electronic Hamiltonian, distinct from the physicist’s operator convention.The construction assumes purely real spatial orbitals.
- Hamiltonian preparation: Real spatial orbitals give the two-electron integral tensor an eight-fold permutation symmetry.The symmetry relates permutations of the tensor indices and supports the matrix-based decomposition.
- Low-rank factorization: Treating the tensor as a matrix indexed by collective pq and rs indices enables eigendecomposition into eigenvalues and eigenvectors.The eigenvalues are denoted w_l, while the resulting eigenvectors generate one-body coefficient structures.
- Basis rotations: The resulting one-body operators are diagonalized in rotated single-particle bases using unitary operators U_l.The basis changes are obtained from eigenvalues of the coefficient tensors and the resulting one-body coefficients.
- Basis rotations: The factorized Hamiltonian absorbs the w_l contributions into g^(l) and diagonalized one-body terms.This produces the representation used for basis-rotation measurement grouping.
Appendix D: Description of Data
The appendix provides raw numerical-calculation data in CSV format for the listed systems. Rows represent systems, while columns report estimator variances and ancillary data such as system energies.
- The appendix includes raw data generated through numerical calculations that does not already appear in the manuscript’s tables.The data is supplied in a CSV file.
- Each CSV row corresponds to one of the systems listed in Table II.The appendix identifies these systems as matching those in the main-text Table II.
- Each column records a different estimator’s variance or ancillary data, with energies in Eh and variances in E2.Ancillary data includes the energy of the system.