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Measuring all compatible operators in one series of a single-qubit measurements using unitary transformations

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arxiv 1907.09386 v4 pith:O3HYUVHN submitted 2019-07-22 quant-ph physics.chem-ph

classification quant-phphysics.chem-ph
keywords groupsoperatorstermscommutinghamiltoniannumberproblemquantum
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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 few 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. It was also shown that this gain scales at least linearly with the number of qubits.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Predicting Features of Quantum Systems from Very Few Measurements

    quant-ph 2019-08 accept novelty 8.0 of 10

    Random Clifford measurements produce a classical shadow of a quantum state that predicts M linear features using only O(log M) measurements, independent of system size, with a matching lower bound.

  2. Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

    quant-ph 2019-08 conditional novelty 7.0 of 10

    All elements of a qubit k-RDM can be measured with O(3^k log^{k-1} N) circuits, and all elements of a fermionic 2-RDM with O(N^2) circuits, matching a new Ω(N^2) lower bound for Clifford measurements.

  3. Variational Quantum Algorithm for Non-equilibrium Steady States

    quant-ph 2019-08 conditional novelty 6.0 of 10

    dVQE variationally computes non-equilibrium steady states of open quantum systems by minimizing the squared Liouvillian over a doubled-qubit ansatz.

  4. Measurement reduction in variational quantum algorithms

    quant-ph 2019-08 conditional novelty 6.0 of 10

    Unitary partitioning can always group molecular electronic-structure Hamiltonian terms into O(N^3) anticommuting sets, reducing the VQE term count by a factor linear in the number of orbitals.

  5. $O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians

    quant-ph 2019-08 conditional novelty 5.0 of 10

    For Jordan-Wigner encoded molecular Hamiltonians, the O(N^4) Pauli terms partition into O(N^3) commuting families of size O(N), cutting VQE measurement cost to O(N^3).

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