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Doubling the size of quantum simulators by entanglement forging

Andrew Eddins, Mario Motta, Tanvi P. Gujarati, Sergey Bravyi, Antonio Mezzacapo, Charles Hadfield, Sarah Sheldon

arXiv:2104.10220v1quant-ph

TL;DR

The paper targets forging a 2N-qubit state from N-qubit components. It describes classically forging entanglement and reports an experiment-count tradeoff against a 2N-qubit device, alongside the reduced qubit requirement per experiment.

Abstract

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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 post-processing allows us to represent ten spin-orbitals on five qubits of an IBM Quantum processor to compute the ground state energy of the water molecule in the most accurate simulation to date. We discuss conditions for applicability of classical entanglement forging and present a roadmap for scaling to larger problems.

SM.1. Decomposition to superpositions of N-qubit bitstrings

This section sets up a decomposition of a 2N-qubit state into superpositions of N-qubit bitstrings. It applies an identity to off-diagonal terms of the state projector to obtain the resulting expression.

SM.1. Decomposition to superpositions of N-qubit bitstrings

SM.2. Entanglement forging in the Heisenberg picture

Heisenberg forging rewrites bipartite Pauli observables as mixtures that can be evaluated on an N-qubit device. Its applicability depends on state and coefficient conditions, and it trades circuit structure against experiment count.

SM.2. Entanglement forging in the Heisenberg picture

  • The ansatz assumes real coefficients and a real unitary, with coefficient vectors that can be sampled and whose pairwise ratios can be computed efficiently.Source: SM.2. Entanglement forging in the… · p. 7
  • The expected value is estimated through N-qubit experiments, with experiment count independent of N and entanglement in the forged state.A typical experiment prepares U†CU|x⟩ and measures each qubit in the standard basis; the circuit uses at most 2(|O1| + |O2|) CNOT gates, and precision ϵ requires a number of experiments proportional to 1/ϵ^2.Source: SM.2. Entanglement forging in the… · pp. 7–8
  • Heisenberg forging rewrites a Pauli observable on two N-qubit registers as a classical mixture of paired N-qubit Clifford observables.Source: SM.2. Entanglement forging in the… · p. 8
  • Compared with Schrödinger forging, Heisenberg forging roughly doubles circuit size but uses shorter circuits than preparing the full 2N-qubit state, while increasing experiments by only a constant factor.Source: SM.2. Entanglement forging in the… · p. 8
  • Using the variational ansatz is justified when the exact ground state has real standard-basis amplitudes and is invariant under exchanging the registers, as for molecular electronic structure with spin-up and spin-down registers.Source: SM.2. Entanglement forging in the… · p. 8
  • The Pauli-observable decomposition is constructed by Clifford transformations that reduce the observables to standard forms.Source: SM.2. Entanglement forging in the… · pp. 9–10

SM.3. State initialization routines

The initialization routines prepare single-qubit product superpositions to represent off-diagonal bitstring terms, allowing entanglement to be classically forged for arbitrary states. Their practicality is constrained by bitstring distance and estimation error.

SM.3. State initialization routines

  • For distinct bitstrings, the routine selects a differing index k; when x_k=1, it swaps x and y and changes p to −p mod 4.It then identifies the other differing indices and the indices where x is 1.Source: SM.3. State initialization routines · p. 11
  • The routine applies X gates on T, Gp on qubit k, and CNOT gates from k to each differing qubit in S.Gp is defined as H, SH, ZH, or SZH, with the Hadamard applied first.Source: SM.3. State initialization routines · p. 11
  • The product state is prepared from |0^N⟩ using Hadamard gates, Pauli X gates, and single-qubit Z rotations by ±πp/2d.Source: SM.3. State initialization routines · p. 12
  • The method classically forges entanglement for arbitrary states by expressing off-diagonal bitstring terms as combinations of product-state terms.The resulting terms can be estimated using N-qubit circuits.Source: SM.3. State initialization routines · p. 12
  • The exponential factor 4^d can cause accuracy loss, so the method is practical only when the Hamming distance d is sufficiently small for contributing bitstring pairs.The error discussion assumes observables with operator norm at most one and N-qubit expectation estimates within additive error ε0.Source: SM.3. State initialization routines · p. 12

SM.4. Connectivity and swaps

The experiment compiles its forged ansatz onto a five-qubit line without additional gate-based swaps. The compilation uses hop-gate variants to match initialization-subcircuit connectivity, while accepting a tradeoff involving circuit errors.

SM.4. Connectivity and swaps

  • The ansatz structure enables compilation onto a five-qubit line without additional gate-based swap operations.Source: SM.4. Connectivity and swaps · p. 12
  • The modified hop gate acts like a hop gate but leaves |11⟩ unchanged, and the figure contrasts the two gate definitions and their compilation.Source: Figure 5 · p. 13, SM.4. Connectivity and swaps · p. 13
  • Unpacked hop gate 2 gives connectivity compatible with two initialization subcircuits; the third requires A-E connectivity and uses the modified hop gate instead.The modified gate removes the swap, providing the needed A-E connection for that subcircuit.Source: SM.4. Connectivity and swaps · p. 13
  • Avoiding swaps reduces total circuit error, but differing compilations can expose circuits to uneven hardware noise and yield nonphysical inferred results.The experiment accepts this tradeoff.Source: SM.4. Connectivity and swaps · p. 13

SM.5. Weighted sampling of circuits: Theory

Entanglement forging estimates a 2N-qubit observable using experiments on N-qubit states, with sampling costs tied to the Schmidt coefficients. Its strongest scalability applies to weakly entangled states.

SM.5. Weighted sampling of circuits: Theory

  • The estimator is unbiased for ⟨ψ|O1 ⊗O2|ψ⟩, with variance bounded by ||µ||2/M.Experiments prepare N-qubit states ρa and measure each subsystem's observable after the corresponding Clifford circuit.Source: SM.5. Weighted sampling of circuits:… · p. 14
  • With 1/M = ϵ2/100, the estimate is within ϵ of the target with probability at least 0.99.The stated confidence follows from the Chebyshev inequality.Source: SM.5. Weighted sampling of circuits:… · p. 14
  • Forging uses roughly 2(||µ||1)^2 times as many experiments as a 2N-qubit device, but requires only N qubits and simpler circuits.The comparison assumes the same target precision and 99% confidence level.Source: SM.5. Weighted sampling of circuits:… · p. 14
  • Forging is mostly useful when the Schmidt-coefficient 1-norm is constant or slowly growing, and an experiment-count cutoff can be imposed as an optimizer constraint.The 1-norm reaches 2^N for a maximally entangled state of N + N qubits.Source: SM.5. Weighted sampling of circuits:… · p. 14

SM.6. Weighted sampling of circuits: Implementation

The implementation approximates weighted circuit sampling by duplicating circuits within submitted jobs, balancing sampling accuracy against job-size overhead.

SM.6. Weighted sampling of circuits: Implementation

  • The copysampling routine includes every circuit, adds floor(wcJ −1) copies, then allocates remaining job slots by residual weight.This approximates the desired distribution when Qiskit samples each listed circuit equally often.Source: SM.6. Weighted sampling of circuits:… · p. 15
  • Weighted sampling is coarse-grained by samples per circuit; the experiment used J ≈800 and s ≈2000 to balance efficiency and job-size overhead.Using s = 1 and J = S would better approximate the target distribution, but job execution time has appreciable overhead scaling with J.Source: SM.6. Weighted sampling of circuits:… · p. 15
  • Preliminary simulations suggest optimized Pauli-aware sampling weights can reduce required samples by more than ∼25%.The proposed weights may incorporate Hamiltonian Pauli-string weights and expectations from the previous VQE iteration.Source: SM.6. Weighted sampling of circuits:… · p. 15

SM.7. Repeated-gate zero-noise extrapolation

Repeated-gate zero-noise extrapolation creates a noise-amplified circuit copy and combines it with the original circuit to estimate a zero-noise result.

SM.7. Repeated-gate zero-noise extrapolation

SM.8. Parallel circuit execution

The implementation runs independent VQE problems in parallel through a software multiplexer, with qubit subsets chosen to reduce crosstalk risk. The accompanying figure describes how truncated Schmidt spaces represent the FCI wavefunction.

SM.8. Parallel circuit execution

  • Parallel runs used qubit sets with no direct connectivity and at least one idle qubit between them to reduce crosstalk risk.Future experiments might also separate qubits by readout-line multiplexing or use readout-crosstalk mitigation.Source: SM.8. Parallel circuit execution · p. 15
  • A software multiplexer merges independent VQE jobs for backend execution, then separates and returns their results to the originating processes.The wrapper can replace the virtual backend with little modification to experiment code.Source: SM.8. Parallel circuit execution · pp. 15–16
  • Figure 6 plots excluded FCI Schmidt-coefficient weight across truncation choices k; values near 0 indicate accurate representation in the truncated space.Its three panels correspond to the geometry sweeps in Figs. 3 and 4, and these FCI calculations do not freeze the oxygen 2p orbital.Source: Figure 6 · p. 16

SM.9. Schmidt coefficients and truncation

The forging demonstration truncates the Schmidt-coefficient space to k = 3 bitstrings, while noiseless simulations compare k = 3 and k = 6. The accuracy impact of k depends on the molecular geometry.

SM.9. Schmidt coefficients and truncation

  • For symmetric stretching, accuracy improves as the molecule dissociates further when k ≥ 6; single-bond cleavage shows a similar but less pronounced change for k ≥ 2.Bond-angle sweeps show no critical k; the FCI calculations include oxygen 2p, increasing the bitstring count from 10 to 15 and leaving a nonzero residual at k = 10.Source: SM.9. Schmidt coefficients and… · p. 16
  • The forging demonstration keeps only the leading k = 3 bitstrings, with noiseless simulations at k = 3 and k = 6 used for comparison.The truncation sets λ_n>k = 0.Source: SM.9. Schmidt coefficients and… · p. 16
  • The accuracy attainable at a chosen k depends on the geometry, with distinct behavior for symmetric stretching, single-bond cleavage, and bond-angle sweeps.The FCI calculations do not freeze oxygen 2p; including it raises the allowed bitstrings from 10 to 15, leaving a nonzero residual for k = 10.Source: SM.9. Schmidt coefficients and… · p. 16

SM.10. Updating Schmidt coefficients

The Schmidt coefficients are updated by exact classical minimization using the latest quantum measurement results. This reduces the VQE optimizer's search to the gate parameters, while allowing real-valued coefficients of either sign.

SM.10. Updating Schmidt coefficients

  • Each VQE iteration samples circuits to estimate the expectation values used to compute the energy across Pauli strings and Schmidt-coefficient indices.The energy calculation combines measured expectation values with sums over p, n, m, and the Hamiltonian's weighted Pauli strings.Source: SM.10. Updating Schmidt coefficients · p. 16
  • The classical processor updates λ from the eigenvector of the known symmetric matrix h with the lowest eigenvalue, so the VQE optimizer searches only over E(ϕ).This removes the Schmidt coefficients from the optimizer's search space.Source: SM.10. Updating Schmidt coefficients · p. 16
  • Allowing Schmidt coefficients to take any real value, rather than requiring non-negative values, facilitates exact minimization and increases ansatz flexibility for fixed U and V.The main-text empirical λ values are shown as absolute values for simplicity.Source: SM.10. Updating Schmidt coefficients · p. 16

SM.11. Example classical simulation with a larger ansatz

At nominal equilibrium geometry, a noiseless classical simulation with a larger ansatz reached 1.47 mH discrepancy from the active-space FCI energy. The example uses ten bitstrings and an empirically selected hop-gate sequence.

SM.11. Example classical simulation with a larger ansatz

  • 1.47 mH discrepancy from the active-space FCI value of -75.727775 H was achieved, with a simulated ground-state energy of -75.726303 H.The target discrepancy was 1.6 mH or less; the simulation used a noiseless classical simulator at nominal equilibrium geometry.Source: SM.11. Example classical simulation… · p. 17
  • The larger ansatz includes all 10 bitstrings consistent with assigning 3 electron-pairs to the 5 active-space molecular orbitals.Its hop gates were found through a heuristic empirical search.Source: SM.11. Example classical simulation… · p. 17
  • The listed hop-gate sequence is ad hoc and is presumed not to be optimal in circuit depth.The orbital indices run from 0 to 6, with 0 and 4 frozen.Source: SM.11. Example classical simulation… · p. 17

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