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Unitary partitioning approach to the measurement problem in the Variational Quantum Eigensolver method
Artur F. Izmaylov, Tzu-Ching Yen, Robert A. Lang, Vladyslav Verteletskyi
TL;DR
VQE energy estimation requires separate measurements of Hamiltonian parts, while single-qubit hardware limits simultaneous measurement to qubit-wise-commuting groups. The paper partitions the Hamiltonian into unitary fragments of anti-commuting terms and reports an N-fold reduction in separately measured groups.
Problem
Single-qubit measurements restrict simultaneous Hamiltonian measurements to qubit-wise-commuting groups, and measuring the parts is time-consuming.
Source: §I · p. 1Method
The method partitions the Hamiltonian into unitary fragments of anti-commuting terms and measures each fragment with a single-qubit measurement circuit.
Source: §I · p. 2, B. Unitary Operator Measuring Circuit · p. 3Results
N-fold fewer operators require separate measurements with unitary partitioning on the tested molecular electronic Hamiltonians.
Source: §IV · p. 4, §III · p. 4Takeaways & Limitations
The tested unitary partitioning reduces the number of separate measurements required for VQE electronic-structure calculations.
Source: §IV · p. 4
Abstract
from arXiv · showhide
To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as $N^4$, where $N$ is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anti-commuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity shows an $N$-fold reduction in the number of measurable groups.
I. INTRODUCTION
VQE estimates electronic energies through an iterative quantum-classical process, but measuring the Hamiltonian is time-consuming because hardware supports single-qubit measurements. The paper proposes grouping anti-commuting Pauli terms into unitary fragments as an alternative to qubit-wise-commuting grouping.
I. INTRODUCTION
- VQE iterates quantum preparation and measurement with classical updates until the energy expectation converges, using relatively short-depth circuits to prepare trial wavefunctions.Source: §I · p. 1
- Single-qubit measurements permit simultaneous grouping only for Pauli terms sharing a tensor-product eigenbasis, or qubit-wise commuting terms.Source: §I · p. 1
- The qubit Hamiltonian has O(N^4) terms in a canonical molecular-orbital basis, while plane waves give quadratic scaling with a large prefactor that pushes the crossover beyond current and near-future hardware.The paper identifies the large quadratic-scaling prefactor as the downside of the plane-wave approach.Source: §I · pp. 1–2
- A prior graph-based QWC grouping reduced measurable groups by an average factor of three, while feed-forward grouping lacked an optimality procedure and was not available on mainstream public hardware.Source: §I · p. 2
- The proposed alternative groups anti-commuting Pauli products into unitary fragments measurable in one series of single-qubit measurements.The paper frames optimal fragment grouping as a minimum clique cover problem on an anti-commutativity graph.Source: §I · p. 2
A. Unitary Partitioning
The section constructs a Hamiltonian representation using unitary fragments, with each fragment formed from anti-commuting Pauli terms. It formulates the grouping as a minimum clique cover problem, while using polynomial heuristics because that problem is NP-hard.
A. Unitary Partitioning
- Each fragment is a normalized linear combination of mutually anti-commuting Pauli terms, which satisfies the stated conditions for representing it as a unitary operator.For real Hamiltonian coefficients, the first two unitarity conditions can be satisfied by renormalization; anti-commutation supplies the third condition.Source: A. Unitary Partitioning · pp. 2–3
- The anti-commutativity graph connects Pauli-word vertices that anti-commute, and its minimum clique cover gives the minimum number of unitary fragments.Source: A. Unitary Partitioning · pp. 2–3
- Because minimum clique cover is NP-hard, the authors use polynomial heuristic algorithms rather than solving it with exponential scaling.Source: A. Unitary Partitioning · p. 3
B. Unitary Operator Measuring Circuit
The circuit estimates the real part of overlaps between the initial state and states generated by controlled unitary transformations, then uses these estimates to obtain the energy. The implementation uses an ancilla and repeated single-qubit measurements.
B. Unitary Operator Measuring Circuit
- The energy estimate is expressed as a sum of weighted real overlaps, with each component directly connected to that estimate.Source: B. Unitary Operator Measuring Circuit · p. 3
- The circuit estimates Re⟨0̄|Φ_n⟩ using an ancilla initialized with the target register, a Hadamard gate, and a controlled unitary transformation.The swap test is avoided because it yields the absolute value of the overlap rather than its real part.Source: B. Unitary Operator Measuring Circuit · p. 3
- Figure 1's overlap-measurement circuit requires N + 1 qubits and M series of single-qubit Z measurements.Source: Figure 1 · p. 3
- The transformed unitary can require 2L − 1 Pauli-word entanglers, giving a worst-case cost that scales as O(NL).Each entangler can be implemented using O(N) CNOT gates and single-qubit rotations.Source: B. Unitary Operator Measuring Circuit · p. 3
- Controlled implementation replaces one-qubit operators with controlled-U gates and decomposes two-qubit operators using Toffoli and controlled-U gates.The controlled transformation is not asymptotically more expensive than its uncontrolled counterpart.Source: B. Unitary Operator Measuring Circuit · pp. 3–4
D. Application to the projection formalism
Symmetry projectors can be represented as linear combinations of unitary operators and incorporated into expectation values of the projected Hamiltonian. This can increase measurement terms while reducing the complexity of preparing the state by satisfying symmetry requirements by construction.
D. Application to the projection formalism
- Symmetry projectors can be expanded into unitary operators and incorporated into expectation values of the projected Hamiltonian.Source: D. Application to the projection… · p. 4
- Although the projector expansion increases the number of measurement terms, it can reduce state-preparation complexity by satisfying symmetry requirements by construction.The projector and Hamiltonian expansions can be combined because products of unitary operators are unitary.Source: D. Application to the projection… · p. 4
III. NUMERICAL STUDIES AND DISCUSSION
Numerical studies compare unitary partitioning with qubit-wise commutativity grouping on molecular electronic Hamiltonians. The reported scaling is N^4 for Hamiltonian terms and N^3 for unitary fragments, corresponding to an N-fold reduction in the number of terms to measure.
III. NUMERICAL STUDIES AND DISCUSSION
- The numerical assessment uses several molecular electronic Hamiltonians, with the paper's broader assessment covering up to fifty thousand Hamiltonian terms.The latter scope is stated in the paper's overview of the study.Source: §III · p. 4
- Among polynomial heuristics, RLF is best in computational time and produces 20% fewer cliques than the next-best algorithm.The authors recommend RLF for larger systems and apply it to them.Source: §III · p. 4
- BKT performs best for the first three Table I systems, but its exponential computational scaling prevents its use on larger systems.Source: §III · p. 4
- N^4 Hamiltonian-term scaling versus N^3 unitary-fragment scaling yields an N-fold reduction in the number of terms to measure.The scaling fits use systems from Table II; Figure 2 plots the total terms and unitary groups against qubit count.Source: §III · p. 4, Figure 2 · p. 4
IV. CONCLUSIONS
The paper partitions qubit Hamiltonians into unitary groups using an anti-commutativity graph and reports an N-fold reduction in separately measured operators. It also identifies circuit-depth and ancilla costs, while noting symmetry projections can be incorporated as linear combinations of unitaries.
IV. CONCLUSIONS
- An N-fold reduction in separately measured operators was found when testing unitary partitioning on molecular electronic Hamiltonians.The scheme increases circuit depth and introduces an additional ancilla qubit.Source: §IV · pp. 4–5
- For a group of L anti-commuting terms and a trial state prepared with K entanglers, the new circuit requires at least 2K + 2L − 1 entanglers.The scheme also introduces an additional ancilla qubit.Source: §IV · p. 5
- The RLF heuristic produced the lowest number of unitary groups among the tested polynomial algorithms, with similar compression rates for JW- and BK-transformed Hamiltonians.The partitioning uses an anti-commutativity graph whose vertices are Pauli words and whose edges connect anti-commuting terms; groups correspond to cliques, and minimizing their number is a minimum clique cover problem.Source: §IV · p. 5
- Symmetry projections can be incorporated straightforwardly because they can be represented as linear combinations of unitary operators.Source: §IV · p. 5
NOTE ADDED:
After the manuscript was submitted to arXiv, several new proposals addressing the measurement problem appeared within a week or two.
NOTE ADDED:
- Several new proposals addressing the measurement problem appeared within a week or two after the manuscript was submitted to arXiv.Source: NOTE ADDED · p. 5
HAMILTONIAN GENERATION
The molecular Hamiltonians were generated using specified molecular geometries, basis sets, and fermion-to-qubit transformations. The resulting systems include H2, LiH, and H2O Hamiltonians of varying qubit counts.
H2
- For H2 at R(H-H) = 1.5 Å, canonical RHF integrals were transformed using Bravyi–Kitaev, with spin-orbitals ordered α, β, α, ....Source: HAMILTONIAN GENERATION · p. 5
LiH
- The LiH Hamiltonian at R(Li-H) = 3.2 Å has 6 qubits and 118 Pauli words; stationary qubits define sectors, including a 4-qubit ground-state sector with 100 Pauli words.The ground-state sector is z3 = −1, z6 = 1.Source: HAMILTONIAN GENERATION · pp. 5–6
H2O
- For H2O, 6- and 26-qubit Hamiltonians used the 6-31G basis, while a 14-qubit Hamiltonian used STO-3G; the geometry was R(O-H) = 0.75 Å and ∠HOH = 107.6°.The 14- and 26-qubit Hamiltonians used JW and BK transformations without modifications.Source: HAMILTONIAN GENERATION · p. 6