Sequency Hierarchy Truncation (SeqHT) for Adiabatic State Preparation and Time Evolution in Quantum Simulations

TL;DR

This work introduces Sequency Hierarchy Truncation (SeqHT), a convergent truncation scheme based on sequency in the Walsh-Hadamard basis to reduce quantum resources for state preparation and time evolution in quantum simulations. By deriving upper bounds on sequency coefficients for polynomial interactions like $λφ^4$, SeqHT enables principled truncation with controllable error, demonstrated through adiabatic state preparation of a $λφ^4$ theory on a 5-qubit site with $ν_{cut}=14$, achieving substantial circuit-depth reductions. The approach yields high fidelities and improved observables on IBM’s quantum hardware, aided by error-mitigation techniques, and shows robust convergence in indicators such as magic for digitized wavefunctions. SeqHT is poised to lower resource barriers in simulations with hierarchies of length scales, potentially extending to lattice gauge theories and beyond, via hybrid quantum-classical loops and adaptable basis choices.

Abstract

We introduce the Sequency Hierarchy Truncation (SeqHT) scheme for reducing the resources required for state preparation and time evolution in quantum simulations, based upon a truncation in sequency. For the $λφ^4$ interaction in scalar field theory, or any interaction with a polynomial expansion, upper bounds on the contributions of operators of a given sequency are derived. For the systems we have examined, observables computed in sequency-truncated wavefunctions, including quantum correlations as measured by magic, are found to step-wise converge to their exact values with increasing cutoff sequency. The utility of SeqHT is demonstrated in the adiabatic state preparation of the $λφ^4$ anharmonic oscillator ground state using IBM's quantum computer ${\textit ibm\_sherbrooke}$. Using SeqHT, the depth of the required quantum circuits is reduced by $\sim 30\%$, leading to significantly improved determinations of observables in the quantum simulations. More generally, SeqHT is expected to lead to a reduction in required resources for quantum simulations of systems with a hierarchy of length scales.

Figures (16)

• Figure 1: Example circuits for Walsh basis operators $\hat{\mathcal{O}}_{24}$ (left) and $\hat{\mathcal{O}}_{10}$ (right) where $24$ and $10$ are the sequency indices. Note that $q_1$ is the most significant qubit and $q_5$ is the least; $R_z$ is a single-qubit rotation gate about the Z axis.
• Figure 2: The coefficients of Pauli strings contributing to the $\hat{\phi}^4$ operator, which is digitized with increasing $n_q$. Each Pauli string has a well-defined sequency, and operators with sequency below $40$ are displayed. (Note that for $n_q=5$, $\hat{\phi}^4$ only decomposes into operators with sequency below $32$.) The coefficients are calculated by $Tr(\hat{\phi}^{4\dag} \hat{\mathcal{O}}_\nu)$. Values for the continuum limit are connected by a dot-dashed line for display purposes. The numerical values used in the subsequent analysis implement a $\nu_{\rm cut}= 14$ sequency truncation, and the values of the results displayed in this figure can be found in Table \ref{['tab:coefficients']}.
• Figure 3: The left panel shows the eigenvalues of full $\phi^4$ theory Hamiltonian and the SeqHT Hamiltonian, both of which are digitized on $n_q = 5$ with $\phi_{\rm max} = 4$ and $\lambda = 10$. The right panel shows the percentage differences between the two sets of eigenvalues. Numerical values for the results displayed in this figure can be found in Table \ref{['tab:eigenvalues']}.
• Figure 4: Amplitudes of the initial ground state, the target ground state of the interacting theory, and the state prepared via the SeqHT ASP procedure which uses five adiabatic steps (not trotterized), with total time $t=5$. In the left panel, the model uses $\lambda = 10$ and achieves a fidelity of $0.9999$. The prepared state is sufficiently close to the target state that the curves essentially coincide. In the right panel, $\lambda = 60$ and fidelity $0.9978$. The $\phi^4$ theory Hamiltonian is digitized onto five qubits ($n_q=5$) with a $\phi_{\rm max} = 4$. Numerical values for the results displayed in this figure can be found in Table \ref{['tab:gs_5qubits']}.
• Figure 5: The fidelity of the $\lambda\phi^4$ ground-state wavefunction prepared with complete adiabatic evolution (blue circles) and with SeqHT procedure (orange crosses). The Hamiltonian is digitized on five qubits with $\phi_{\rm max} = 4$ and $\lambda = 10$. Size of each time step is fixed at $\delta t = 0.1$ and number of steps is taken from $0$ to $80$ for a total time from $0$ to $8$. The horizontal line indicates the initial overlap ($0.9729$) of the free theory and the interacting theory ground states. Numerical values in this plot can be found in Table \ref{['tab:fidelity_time']}.