We propose a method to sequentially optimize arbitrary single-qubit gates in parametrized quantum circuits for simulating real- and imaginary-time evolution. The method utilizes full degrees of freedom of single-qubit gates and therefore can potentially obtain better performance. Specifically, it simultaneously optimizes both the axis and the angle of a single-qubit gate, while the known methods either optimize the angle with the axis fixed, or vice versa. It generalizes the known methods and utilizes sinusoidal cost functions parametrized by the axis and angle of rotation. Furthermore, we demonstrate how it can be extended to optimize a set of parametrized two-qubit gates with excitation-conservation constraints, which includes the HOP and the reconfigurable beam-splitter gates. We perform numerical experiments showing the power of the proposed method to find ground states of typical Hamiltonians with quantum imaginary-time evolution using parametrized quantum circuits. In addition, we show the method can be applied to real-time evolution and discuss the tradeoff between its simulation accuracy and hardware efficiency.
We propose a method to sequentially optimize arbitrary single-qubit gates in parametrized quantum circuits for simulating real- and imaginary-time evolution. The method utilizes full degrees of freedom of single-qubit gates and therefore can potentially obtain better performance. Specifically, it simultaneously optimizes both the axis and the angle of a single-qubit gate, while the known methods either optimize the angle with the axis fixed, or vice versa. It generalizes the known methods and utilizes sinusoidal cost functions parametrized by the axis and angle of rotation. Furthermore, we demonstrate how it can be extended to optimize a set of parametrized two-qubit gates with excitation-conservation constraints, which includes the HOP and the reconfigurable beam-splitter gates. We perform numerical experiments showing the power of the proposed method to find ground states of typical Hamiltonians with quantum imaginary-time evolution using parametrized quantum circuits. In addition, we show the method can be applied to real-time evolution and discuss the tradeoff between its simulation accuracy and hardware efficiency.
We propose a novel method to sequentially optimize arbitrary single-qubit gates in parameterized quantum circuits for simulating real and imaginary time evolution. Unlike previously-known methods, the proposed method fully utilizes all parameters of single-qubit gates and therefore can potentially obtain better performance. Specifically, the proposed method simultaneously optimizes both the axis and the angle of a single-qubit gate, while the known methods either optimize the angle with the axis fixed, or vice versa. The proposed method is shown to generalize the known methods and results in a sinusoidal cost function parameterized by the axis and angle of rotation. Furthermore, we demonstrate how the proposed method can be extended to optimize a set of parameterized two-qubit gates with excitation-conservation constraints, which includes the Hop and the Reconfigurable Beam Splitter gates. We perform numerical experiments showing the power of the proposed method to find ground states of typical Hamiltonians with quantum imaginary time evolution using parameterized quantum circuits.
We propose two-gate extensions of the sequential single-qubit optimizers, Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), termed Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), respectively. In contrast to Fraxis and FQS, which update one single-qubit gate at a time via quadratic local cost function and matrix diagonalization, TGF and TGFQS optimize two parameterized single-qubit gates simultaneously by constructing an exact quartic local cost function and optimizing it using classical optimizers. We further investigate how different gate pairing strategies affect optimization performance. Using numerical experiments on spin Hamiltonians, molecular Hamiltonians, and quantum state preparation tasks, we find that TGF and TGFQS frequently achieve a lower final relative error to the ground state energy or infidelity than their single gate counterparts. We observe that the random and half-shifted gate pairing strategies for TGF and TGFQS perform best in many of the tested settings. In the additional finite-shot experiments on Fermi-Hubbard and transverse-field Ising model Hamiltonians, the best gate pairing strategies retain their advantage across the tested shot counts in shallow circuits. These improvements come at the cost of increased circuit evaluations per gate update, highlighting a trade-off between the power of local optimization and measurement overhead.
Variational quantum algorithms, which utilize Parametrized Quantum Circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parametrized rotational angles of single-qubit gates around predetermined set of axes, and two-qubit entangling gates, such as CNOT gates. We propose a method to construct a PQC by continuous parametrization of both the angles and the axes of its single-qubit rotation gates. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed toπ. We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback-Leibler (KL) divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Hamiltonians for condensed matter physics, quantum chemistry and combinatorial optimization. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.
Parameterized quantum circuits (PQCs), which are essential for variational quantum algorithms, have conventionally been optimized by parameterized rotational angles of single-qubit gates around predetermined set of axes. We propose a new method to optimize a PQC by continuous parameterization of both the angles and the axes of its single-qubit rotations. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed to π. We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback-Leibler (KL) divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Heisenberg models and molecular Hamiltoni- ans. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.
We propose two-gate extensions of the sequential single-qubit optimizers, Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), termed Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), respectively. In contrast to Fraxis and FQS, which update one single-qubit gate at a time via a quadratic local cost function and matrix diagonalization, TGF and TGFQS optimize two parameterized single-qubit gates simultaneously by constructing an exact quartic local cost function and optimizing it using classical optimizers. We further investigate how different gate pairing strategies affect optimization performance. Using numerical experiments on lattice Hamiltonians, molecular Hamiltonians, and quantum state preparation tasks, we find that TGF and TGFQS frequently achieve a lower final relative error to the ground state energy or infidelity than their single-gate counterparts. We observe that the random and half-shifted gate pairing strategies for TGF and TGFQS perform best across many of the tested settings. In the additional finite-shot experiments on Fermi–Hubbard and transverse-field Ising model Hamiltonians, the best gate pairing strategies retain their advantage across the tested shot counts in shallow circuits. These improvements come at the cost of increased circuit evaluations per gate update, highlighting a trade-off between the power of local optimization and measurement overhead.
Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.