Doubling the Size of Quantum Simulators by Entanglement Forging

Quantum computers are promising for simulations of chemical and physical systems, but the limited capabilities of today’s quantum processors permit only small, and often approximate, simulations. Here we present a method, classical entanglement forging, that harnesses classical resources to capture quantum correlations and double the size of the system that can be simulated on quantum hardware. Shifting some of the computation to classical postprocessing allows us to represent ten spin orbitals of the water molecule on five qubits of an IBM Quantum processor in the most accurate variational simulation of the ${\mathrm{H}}_2\mathrm{O}$ ground-state energy using quantum hardware to date. We discuss conditions for applicability of classical entanglement forging and present a roadmap for scaling to larger problems.

Popular Summary

The ability to accurately simulate quantum systems of increasing complexity is important for discovering new chemicals and materials. However, the task is challenging: classical computations are frustrated by the intrinsic inefficiency of classically representing a quantum state, while quantum computations are restricted to small simulations due to the limited quantity and quality of available quantum bits. One promising approach is to maximize overall computational power by judiciously assigning different pieces of the simulated quantum state to classical and quantum subroutines. Here we present a method, “classically forged entanglement,” in which a quantum processor simulates individually the two halves of a quantum state, while a classical processor simulates their entanglement, enabling $N$ qubits to simulate a $2N$-qubit system.

We demonstrate the method in a simulation of the ground state of the water molecule, using five qubits to simulate ten spin orbitals. The quantum processor repeatedly prepares and measures a state representing either the spin-up or spin-down electrons, and the results are combined with classical parameters defining the entanglement to compute the energy of the state. Critically, spin-up and -down electrons are only weakly entangled in the ground state of water, and we use the simple structure of this entanglement to greatly improve performance of the classical computation. As weak entanglement is a common property of many molecular ground states, we expect this procedure to significantly expand the class of chemical and physical problems of interest that can be studied on quantum simulators in the near term and beyond.

Figure 1

Classically forged entanglement. (a) A state $|\mathrm{\Psi }⟩$ of a bipartite quantum system, labeled with arrows alluding to spin polarization, can be defined by gates $E$, $U$, and $V$, where $E$ outputs a combination of bit-string states $|b_n⟩|b_n⟩$. (b) A two-qubit entangled state can be rewritten using one-qubit superposition states. Changing labels $0,1\to b_n,b_m$ gives a transformation acting on components of the $2N$-qubit state. (c) $|\mathrm{\Psi }⟩$ can be reconstructed from $N$-qubit circuits initialized as bit strings and pairwise superpositions thereof. Circuits associated with small ${\lambda }_n{\lambda }_m$ can be estimated adequately from few samples. (d) Rapid (slow) decay of the leading Schmidt coefficients in the decomposition of a molecular ground state signals weak (strong) entanglement between spin-up and spin-down particles.

Figure 2

Experimental ansatz structure. (a) The five active molecular orbitals are encoded in a line of qubits via the Jordan-Wigner mapping. The qubits are prepared in either a bit-string state, or a superposition of two bit strings (shown), and then acted upon by parameterized hop gates (b). In order to evaluate the Pauli strings of the Hamiltonian, each state is prepared many times, and rotated appropriately prior to measurement in the computational basis. (c) Parameterized molecular geometry from which molecular orbitals are determined. (d) Device map of ibmq_dublin, with a highlighted example of how two geometries may be solved in parallel.

Figure 3

VQE energies. Ground-state energies computed while varying (a) both O—H bond lengths, (b) a single O—H bond length, and (c) the H—O—H angle $\theta$. In the upper plots, VQE results using the entanglement-forging ansatz on ibmq_dublin appear as filled blue circles, alongside curves indicating the classically computed Hartree-Fock (HF) and full configuration interaction (FCI) values for the active space. Unfilled shapes indicate results from running VQE on a noiseless classical simulator using the same three-bit-string ansatz used on ibmq_dublin ($k=3$), and using a larger six-bit-string ansatz ($k=6$). Lower plots indicate absolute errors, $|E-E_{\mathrm{FCI}}|$. Error bars are one standard deviation produced by bootstrapping measurement histograms, representing precision but not systematic error or drift.

Figure 4

Entanglement structure. Schmidt coefficients ${\lambda }_n$ obtained from VQE run on ibmq_dublin (solid circles) and on a noiseless classical simulator (empty circles) while varying molecular geometry as in Fig. 3. The largest component is the Hartree-Fock amplitude (${\lambda }_1$, red); the two remaining values are sorted and plotted in blue and green. (a) Bit strings beyond Hartree Fock become increasingly significant as the molecular bonds are stretched. (b) Removing a single hydrogen produces a ground-state wave function dominated by two bit strings. (c) All choices of H—O—H bond angle yielded ground states characterized by weak entanglement.

Figure 5

Compilation onto a line of qubits. (a) Definition of the hop gate (green diamonds) in terms of a swap followed by standard gate operations, the latter abbreviated by the gate with green circles. (b) Definition of the modified hop gate, equivalent to a hop gate except that it leaves the $|11⟩$ state unchanged. (c) Initial compilation steps for the hop gates used in the experiment. (d) Likewise unpacking hop-gate 2 per its definition leads to the circuit in the upper right, the connectivity of which is compatible with the two state-initialization subcircuits shown (purple rectangles). (e) Alternatively, replacing hop-gate 2 with a modified hop gate provides a connectivity solution for the state-initialization subcircuit shown in this panel.

Figure 6

Accuracy of truncated Schmidt decompositions. The three panels correspond to the three sweeps of molecular geometries as in Figs. 3, 4. For each choice of bit-string truncation $k$, we plot one minus the projection of the FCI wave function into the space defined by the leading $k$ bit strings. In other words, we plot the sum of squares of the FCI Schmidt coefficients excluded by the truncation. Values near 1 (0) indicate that the FCI wave function cannot (can) be accurately represented in the truncated space. Unlike in the rest of the text, these FCI calculations do not freeze the oxygen $2p$ orbital.