Schrödinger-Heisenberg Variational Quantum Algorithms

TL;DR

This work proposes a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA), which enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally.

Abstract

Recent breakthroughs have opened the possibility to intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the extremely high accuracy needed to surpass classical computers poses a critical demand to the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1-1%. Here, by incorporating a virtual Heisenberg circuit, which acts effectively on the measurement observables, to a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms to resolve this problem. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be efficiently and classically implemented according to the Gottesman-Knill theorem. Yet, it greatly enlarges the state expressivity, realizing much larger unitary t-designs. Our method enables accurate quantum simulation and computation that otherwise is only achievable with much deeper and more accurate circuits conventionally. This has been verified in our numerical experiments for a better approximation of random states and a higher-fidelity solution to the ground state energy of the XXZ model. Together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.

Figures (9)

• Figure 1: SH-VQE. (a): The SH-VQE circuit. The circuit is composed of the Schrödinger circuit $U$ and the Heisenberg circuit $T$, where $U$ is the local unitary circuit running on real quantum computers and $T$ is the virtual circuit acted on the Hamiltonian consisting of two parts, the Clifford part, and the single qubit layer. The architecture we use for $U$ throughout this work is layers of parallel 2-qubit gates, which has a well-defined light cone that constrains the propagation of correlations and entanglements. (b): Improvements of SH-VQE. By adding the virtual circuit, $T U\left|0^{\otimes n}\right\rangle$ is able to explore more of the Hilbert space compared with $U\left|0^{\otimes n}\right\rangle$ in conventional VQE and the trainable Hilbert space is much larger than the conventional VQE. (c): Algorithm structure comparison between VQE and SH-VQE. The transformed Hamiltonian $H_{T}$ replaces $H$ in SH-VQE. We update parameters in both $U$ and $T$ to minimize the expectation value of $H_{T}$.
• Figure 2: SH-VQE expressivity. (a): Relationship between expressivity and the $t$-design. We show the point distribution on the Bloch sphere of different design orders $t=3,6,9,12$. (b): Comparison of the expressivity measure $\Delta_{t}$ between VQE and SH-VQE. The structure of the Clifford part is formed by 500 randomly picked basic Clifford gates in the set $\{\mathrm{H}, \mathrm{S}, \mathrm{CNOT}\}$. The other parts including the two-qubit blocks in Schrödinger LUC and gates in the Heisenberg single qubit layer are random gates drawn from the Haar measure. The zero-depth setting in SH-VQE can be understood as the performance of the Clifford circuit. Since Clifford circuit can generate 3-design, $\Delta_{t}$ approaches 0 for $t=3$ whereas below 0 in other cases. Schrödinger circuit of depth greater than 6 combined with the Heisenberg circuit is believed to generate the maximally scrambled states since values of $\Delta_{t}$ from $t=3$ to $t=$ 12 are all zero liu2018entanglement.
• Figure 3: Searching the Clifford circuit for the XXZ model. (a): The 4 elementary graphs and their corresponding code strings for $n=8$ TI graphs. (b): Upper Panel: Minimizing the cost function to search for the best graph. Parameters contain both gate parameters and probability parameters. The cost function is the sum of the Hamiltonian expectation values of circuits sampled from $\vec{\alpha}$ under the same gate parameters. The number of samples at each iteration is 800. Lower Panel: Probabilities of all 16 graphs as functions of iteration times. All graphs have the same probabilities at the beginning. The probability of the fully connected graph '1111' becomes 1 as the iteration times grow. The optimization algorithm used for circuit structure searching is adam-SPSA arouri2020accelerated. (c): Direct comparisons between different graphs. The dashed line is the result of the graph type: '0000' i.e. without the Heisenberg circuit. The fully connected graph is indeed the best choice. There exist graphs that have worse performance than '0000'. (d): Comparison of solved ground state fidelities. We use VQE and SH-VQE to solve 8, 10, 12,14, and 16-qubit XXZ models. Fully connected graphs are used as the Clifford layer. VQE and SH-VQE of the same Schrödinger circuit depth share the same color. Each point is the best result obtained from 20 sets of random initial parameters.
• Figure 4: Comparisons of SH-VQE and VQE on solving small molecules. Using VQE and SH-VQE to solve the 8-qubit Hamiltonian of the $\mathrm{H}_{4}$ molecule of the bond distance 1.0 Angstrom and the 10-qubit Hamiltonian of the $\mathrm{H}_{2} \mathrm{O}$ molecule of the bond distance 1.5 Angstrom. The binary string around the SH-VQE data label the searched optimal graph circuit. (a): Solved energy as a function of Schrödinger circuit depth. (b): Absolute energy differences as functions of Schrödinger circuit depth.
• Figure B.1: Structure of parametrized circuits used when solving XXZ models. Two qubit gates are fixed $\mathrm{CZ}$ gates while each single-qubit gate is parametrized as $\exp \left(-i \theta_{\mathrm{x}} \sigma_{\mathrm{x}}\right) \exp \left(-\mathrm{i} \theta_{\mathrm{y}} \sigma_{\mathrm{y}}\right) \exp \left(-\mathrm{i} \theta_{\mathrm{z}} \sigma_{\mathrm{z}}\right)$. The Clifford part is composed of only CZ gates.