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arXiv:1907.09386 · Journal of Chemical Theory and Computation

Measuring All Compatible Operators in One Series of Single-Qubit Measurements Using Unitary Transformations

Tzu-Ching Yen, Vladyslav Verteletskyi, Artur F. Izmaylov

Published Mar 9, 2020• 178 citations
arXivHTMLPDF

Abstract

The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or compatible operators simultaneously. Unfortunately, the current hardware permits measuring only a much more limited subset of operators that share a common tensor product eigen-basis. We introduce unitary transformations that transform any fully commuting group of operators to a group that can be measured on current hardware. These unitary operations can be encoded as a sequence of Clifford gates and let us not only measure much larger groups of terms but also to obtain these groups efficiently on a classical computer. The problem of finding the minimum number of fully commuting groups of terms covering the whole Hamiltonian is found to be equivalent to the minimum clique cover problem for a graph representing Hamiltonian terms as vertices and commutativity between them as edges. Tested on a set of molecular electronic Hamiltonians with up to 50 thousand terms, the introduced technique allows for the reduction of the number of separately measurable operator groups down to few hundreds, thus achieving up to 2 orders of magnitude reduction. Based on the test set results, the obtained gain scales at least linearly with the number of qubits.

DOI
10.1021/acs.jctc.0c00008

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The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or compatible operators simultaneously. Unfortunately, the current hardware permits measuring only a much more limited subset of operators that share a common tensor product eigen-basis. We introduce unitary transformations that transform any fully commuting group of operators to a group that can be measured on current hardware. These unitary operations can be encoded as a sequence of Clifford gates and let us not only measure much larger groups of terms but also to obtain these groups efficiently on a classical computer. The problem of finding the minimum number of fully commuting groups of terms covering the whole Hamiltonian is found to be equivalent to the minimum clique cover problem for a graph representing Hamiltonian terms as vertices and commutativity between them as edges. Tested on a set of molecular electronic Hamiltonians with up to 50 thousand terms, the introduced technique allows for the reduction of the number of separately measurable operator groups down to few hundreds, thus achieving up to 2 orders of magnitude reduction. Based on the test set results, the obtained gain scales at least linearly with the number of qubits.
Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as O(N4) with the size of the system, N. To optimize the VQE measurement, one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard, but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times less compared to the total number of terms in the considered Hamiltonians.
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 that 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, thereby putting 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 anticommuting 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 is consistent with an N -fold reduction in the number of measurable groups.
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule, this increases the uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: 1) grouping terms into parts whose eigenstates have a single-qubit product structure, and 2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in several-fold reduction of separately measured terms are illustrated on the H2 and LiH problems.
Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10--30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual-basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.
The Variational Quantum Eigensolver (VQE) requires a large number of measurements to evaluate molecular Hamiltonians. Expressing a molecular Hamiltonian as a linear combination of Pauli strings creates a measurement bottleneck: non-commuting Pauli strings cannot be measured simultaneously. Consequently, mutually commuting Pauli strings must be grouped and measured together to minimise the number of quantum-state preparations. This task maps to the minimum clique cover problem on a commutativity graph, an NP-hard problem typically addressed using classical heuristics. Although an Ising-model formulation has recently been explored on fully connected CMOS Ising machines and demonstrated on physical quantum annealers only at small scale, the embedding cost that governs its behaviour on hardware with sparse connectivity, where each logical variable must be represented by a chain of physical qubits, has not been characterised. In this work, we formulate the Pauli-grouping problem as a standard QUBO colouring model and study its scalability on a D-Wave quantum annealer. Across a series of molecular systems and for both qubit-wise and full commutativity, we quantify the growth in the number of logical variables and the QUBO interaction density, characterise the physical-qubit and chain-length overhead required for embedding on the Zephyr topology, and compare the annealer's time-to-solution and solution quality with those of heuristic classical baselines. This analysis identifies the threshold of molecular complexity beyond which hardware connectivity prevents viable embedding, and shows that a second, practical limit is reached earlier. The threshold therefore measures how far current annealers are from the regime in which pre-optimising a VQE measurement scheme on hardware would be worth considering.
Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, one of the strategies is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate estimation of the observable's expectation value. We consider three previously studied directions that minimize the number of measurements: (1) grouping commuting operators using the greedy approach, (2) involving non-local unitary transformations for measuring, and (3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.