Numerical Experiments with Parameter Setting of Trotterized Quantum Phase Estimation for Quantum Hamiltonian Ground State Computation

TL;DR

One of the notable properties of QPE sampling of the optimal digitized phase converges to a fixed rate results in strong diminishing returns of optimal phase sampling rates which can occur when the Trotter error is surprisingly high.

Abstract

We numerically investigate quantum circuit elementary-gate level instantiations of the standard Quantum Phase Estimation (QPE) algorithm for the task of computing the ground-state energy of a quantum magnet; the disordered fully-connected quantum Heisenberg spin glass model. We consider (classical simulations of) QPE circuit computations on relatively small quantum Hamiltonians ($3$ qubits) with up to $10$ phase bits of precision, using up to Trotter order $10$. We systematically study the inputs of QPE, specifically time evolution, Trotter order, Trotter steps, and initial state, and illustrate how these inputs practically determine how QPE operates. From this we outline a coherent set of quantum algorithm input and tuning guidelines. One of the notable properties we characterize is that QPE sampling of the optimal digitized phase converges to a fixed rate. This results in strong diminishing returns of optimal phase sampling rates which can occur when the Trotter error is surprisingly high.

Figures (10)

• Figure 1: Quantum circuit diagram schematic of the Quantum Phase Estimation algorithm to compute the minimum eigenvalue of a quantum Hamiltonian, using $3$ qubits for the phase register and $3$ qubits for the unitary evolution register as a small scale example. The unitary evolution register is initialized in the state ${\mathcal{A}}$, followed by evolution by controlled powers of $U$, which in this case are implemented by Trotter-Suzuki (or first order Lie-Trotter) decomposition. The algorithm terminates by performing an inverse quantum Fourier transform (QFT) on the phase qubits, at which point the qubits are measured in the computational $Z$ basis. A single measured bit vector from the algorithm is an estimate of the phase, which can be translated to an estimate of the ground-state energy of a quantum Hamiltonian. Here, $H$ denotes the Hadamard gate.
• Figure 2: Digitization phase readout error. Error (log-scale) with respect to the optimal quantum Hamiltonian ground-state energy as a function of increased bits provided for the digitization approximation of the ground-state energy, using a total time evolution given by Eq. \ref{['equation:heuristic_evolution_time']} and the minimum eigenvalue found by exact diagonalization for Heisenberg spin glass models with up to $10$ qubits. Error is defined as the absolute value of the difference between the ground state energy ($E_0$) and the closest bitstring digitization of that energy. Note that this data is not from a QPE circuit simulation, rather, this is showing the lowest possible error for each number of bits of precision up to $22$.
• Figure 3: Overlap (y-axis) between various easy-to-prepare initial states and the ground-states(s) eigenvector(s) of the quantum Hamiltonian (x-axis). The overlap quantity $\chi$ is defined by Eq. \ref{['equation:degenerate_overlap']}.
• Figure 4: QPE optimal phase sampling (y-axis) as a function of Trotter steps $r$ (x-axis). The time evolution used is the $t_0$ from Eq. \ref{['equation:heuristic_evolution_time']}. At small $r$, we observe clear transitory effects due to the time evolution being approximated poorly. Higher order $k$ at larger $r$ is not reported because of the substantial gate counts required to represent the full QPE circuit, and more specifically the classical circuit simulation of those large circuits. The dashed red line labeled Upper Bound* plots Eq. \ref{['equation:px_upper_bound']}, and the dashed cyan line plots Eq. \ref{['equation:QPE_optimal_phase_sampling_rate']}.
• Figure 5: High resolution search over evolution times. Measuring the optimal phase sampling rate over a gridsearch of evolution times specified by a range starting from the time given by Eq. \ref{['equation:heuristic_evolution_time']} (vertical dashed green line), and then down to $t_0 - 8\cdot \frac{t_0}{2^{m_{prec}}}$. This scaling of the time resolution is because the more phase qubits are used, the shorter the periodicity of the time evolution becomes. The dashed red line labeled Upper Bound* plots Eq. \ref{['equation:px_upper_bound']}, and the dashed cyan line plots Eq. \ref{['equation:QPE_optimal_phase_sampling_rate']} as a function of time. In all four sub-plots the time evolution begins at precisely the same $t_0$, however, where in this periodic time landscape $t_0$ is depends on $m_{prec}$. Note that when $t$ changes, the optimal phase can also change, which means that the y-axis is not necessarily plotting the sampling rate of a single bitstring. The bottom row shows two examples of the transitory effects that occur when the Trotter error is too high, and the therefore the sampling rate has not converged to Eq. \ref{['equation:QPE_optimal_phase_sampling_rate']}.