A central roadblock in the realization of variational quantum eigensolvers on quantum hardware is the high overhead associated with measurement repetitions, which hampers the computation of complex problems, such as the simulation of mid- and large-sized molecules. In this work, we propose a novel measurement protocol which relies on computing an approximation of the Hamiltonian expectation value. The method involves measuring cheap grouped operators directly and estimating the residual elements through iterative measurements of new grouped operators in different bases, with the process being truncated at a certain stage. The measured elements comprehend the operators defined by the Hard-Core Bosonic approximation, which encode electron-pair annihilation and creation operators. These can be easily decomposed into three self-commuting groups which can be measured simultaneously. Applied to molecular systems, the method achieves a reduction of 30% to 80% in the number of measurement and gates depth in the measuring circuits compared to state-of-the-art methods. This provides a scalable and cheap measurement protocol, advancing the application of variational approaches for simulating physical systems.
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.
Estimating the expectation value of a Hamiltonian and performing a Hamiltonian simulation are two fundamental tasks in quantum computation, with their efficiency heavily dependent on the Pauli norm of the Hamiltonian, which sums the absolute values of its Pauli coefficients. In this work, we propose a variational quantum algorithm called variational quantum Hamiltonian engineering (VQHE) to minimize the Pauli norm of a Hamiltonian, thereby reducing the overhead for expectation value estimation and Hamiltonian simulation. We first develop a theoretical framework that transforms the Pauli norm minimization problem into a vector l 1 -norm minimization problem. We then design an appropriate cost function and employ parametrized quantum circuits to variationally minimize this cost function. Numerical experiments demonstrate the effectiveness of VQHE in reducing the Pauli norm for both the Ising Hamiltonian and molecular Hamiltonians. Furthermore, we show that VQHE is compatible with grouping strategies, enabling further reductions in measurement complexity for expectation value estimation. Our results highlight the potential of VQHE to enhance the efficiency of quantum algorithms in near-term quantum devices, offering a promising approach for practical quantum computing applications.
The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the noncommutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators. Published by the American Physical Society 2024
Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.
We proposed a simple strategy to improve the postprocessing overhead of evaluating Hamiltonian expectation values in Variational quantum eigensolvers (VQEs). Observing the fact that for a mutually commuting observable group G in a given Hamiltonian, is fixed for a measurement outcome bit string
b in the corresponding basis, we create a measurement memory (MM) dictionary for every commuting operator group G in a Hamiltonian. Once a measurement outcome bit string
b appears, we store
b and as key and value, and the next time the same bit string appears, we can find from the memory, rather than evaluate it once again. We further analyze the complexity of MM and compare it with commonly employed post-processing procedure, finding that MM is always more efficient in terms of time complexity. We implement this procedure on the task of minimizing a fully connected Ising Hamiltonians up to 20 qubits, and
H2,
H4,
LiH, and
H2O molecular Hamiltonians with different grouping methods. For Ising Hamiltonian, where all
O(N2) terms commute, our method offers an
O(N2) speedup in terms of the percentage of time saved. In the case of molecular Hamiltonians, we achieved over
O(N) percentage time saved, depending on the grouping method.