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Gate-efficient simulation of molecular eigenstates on a quantum computer
Marc Ganzhorn, Daniel J. Egger, Panagiotis Kl. Barkoutsos, Pauline Ollitrault, Gian Salis, Nikolaj Moll, Andreas Fuhrer, Peter Mueller, Stefan Woerner, Ivano Tavernelli, Stefan Filipp
TL;DR
Accurate molecular simulations require circuits short enough to fit within near-term hardware coherence times, while unconstrained VQE circuits can explore exponentially growing Hilbert spaces. This paper implements excitation-preserving exchange gates on superconducting qubits and uses VQE and EOM to calculate molecular hydrogen eigenstates. The reported accuracy remains above chemical accuracy, with tunable-coupler coherence identified as the main limitation.
Problem
Accurate molecular simulations require substantial resources, and unconstrained parameterized VQE circuits access Hilbert spaces whose dimension grows exponentially with the number of qubits.
Source: p. 1Method
The authors implement tunable-amplitude-and-phase exchange-type gates on superconducting qubits for VQE ground-state calculations and use EOM to derive molecular hydrogen excited states.
Source: p. 5Results
Ground- and excited-state energy errors remain above chemical accuracy, and the tunable coupler’s coherence time is identified as the main cause of inaccuracy.
Source: p. 5Takeaways & Limitations
Exchange-type gates preserve excitation number and reduce the VQE entangler to a single gate primitive, supporting efficient molecular ground- and excited-state calculations in this architecture.
Source: p. 5The present computation’s accuracy is limited mainly by the tunable coupler’s coherence time; simulations indicate that 500 ns coherence would enable chemical accuracy in this architecture.
Source: p. 5
Abstract
from arXiv · showhide
A key requirement to perform simulations of large quantum systems on near-term quantum hardware is the design of quantum algorithms with short circuit depth that finish within the available coherence time. A way to stay within the limits of coherence is to reduce the number of gates by implementing a gate set that matches the requirements of the specific algorithm of interest directly in hardware. Here, we show that exchange-type gates are a promising choice for simulating molecular eigenstates on near-term quantum devices since these gates preserve the number of excitations in the system. Complementing the theoretical work by Barkoutsos et al. [PRA 98, 022322 (2018)], we report on the experimental implementation of a variational algorithm on a superconducting qubit platform to compute the eigenstate energies of molecular hydrogen. We utilize a parametrically driven tunable coupler to realize exchange-type gates that are configurable in amplitude and phase on two fixed-frequency superconducting qubits. With gate fidelities around 95% we are able to compute the eigenstates within an accuracy of 50 mHartree on average, a limit set by the coherence time of the tunable coupler.
computer
The paper is by M. Ganzhorn and coauthors at IBM Research GmbH’s Zurich Research Laboratory; it is dated September 14, 2018.
- The paper is dated September 14, 2018, and lists IBM Research GmbH’s Zurich Research Laboratory as the authors’ affiliation.The author list begins with M. Ganzhorn and ends with S. Filipp.Source: computer · p. 9
ESTIMATION OF CIRCUIT DEPTH AND REQUIRED HARDWARE SIZE
The section compares circuit-depth requirements for excitation-conserving and non-excitation-conserving VQE circuits. Exchange-type gates reduce depth, while decomposing them into CNOT-based operations increases depth ninefold.
Heuristic excitation conserving circuits
- Excitation-conserving circuits use blocks of N −1 exchange-type gates, with depth estimated assuming error-free gates and nearest-neighbour connectivity.The estimates use STO3G basis sets, Jordan-Wigner mapping, and effective core potentials for LiH, BeH2, and H2O, reducing their qubit counts by two.Source: Heuristic excitation conserving circuits · pp. 10–11
- Errors in gates or reduced connectivity increase the estimated circuit depth for excitation-conserving circuits.The comparison also estimates runtime using assumed gate times and includes decomposed exchange gates for architectures without native exchange-type gates.Source: Heuristic excitation conserving circuits · p. 11
ESTIMATION OF CIRCUIT DEPTH AND REQUIRED HARDWARE SIZE
- Ninefold greater circuit depth is required when exchange-type gates are decomposed into single- and two-qubit rotations, though the studied circuits remain within the T1 limit.The runtime estimates assume gate times of 200 ns for CNOT gates, 170 ns for exchange-type gates, and 25 ns for single-qubit gates.Source: Heuristic excitation conserving circuits · pp. 11–12
Heuristic non excitation-conserving circuits
- Non-excitation-conserving VQE circuits use blocks of single-qubit rotations and CNOT-based entanglers, with minimum block counts estimated at each molecule’s equilibrium geometry.The estimates assume error-free gates and all-to-all connectivity.Source: Heuristic non excitation-conserving… · pp. 12–13
- Non-excitation-conserving circuits exceed available T1 times for LiH, while BeH2 and H2O did not reach chemical accuracy within 26 blocks.The simulations assume error-free gates and all-to-all connectivity; greater depth would likely be required with gate errors or reduced connectivity.Source: Heuristic non excitation-conserving… · p. 13
COHERENCE MEASUREMENTS AND FLUX NOISE CHARACTERIZATION
The tunable coupler’s coherence is characterized using sideband-driven population transfer, while the measured coherence times also inform an estimate of flux noise. The computational qubits are likewise measured across magnetic flux because they are exposed to flux noise through the coupler architecture.
COHERENCE MEASUREMENTS AND FLUX NOISE CHARACTERIZATION
- The tunable coupler’s population can be measured indirectly through a computational qubit, and a sideband drive transfers population between the qubit and coupler.A full transfer from |1g⟩ to |0e⟩ is achieved at x = π.Source: Tunable coupler transmon · p. 14
- Measured coherence times yield a flux-noise amplitude of 5.4 ± 1.4 · 10−5 Φ0 for a tunable-coupler SQUID loop of 25 × 25 µm2.Source: Tunable coupler transmon · p. 15
- Both computational qubits’ T1, T2, and T ∗ 2 times are measured as functions of magnetic flux.The coupler architecture exposes the qubits to flux noise.Source: Computational transmon qubits · p. 15
RANDOMIZED BENCHMARKING OF ISWAP GATES IN TC ARCHITECTURE
Randomized benchmarking characterizes the iSWAP gate, which is a Clifford gate in this implementation. The measured error per gate agrees with decoherence simulations for longer gates, while short gates show increased error associated with leakage.
RANDOMIZED BENCHMARKING OF ISWAP GATES IN TC ARCHITECTURE
- Randomized benchmarking measures the iSWAP error per gate using two datasets, distinguished by whether qubit 1 or qubit 2 is measured.Source: RANDOMIZED BENCHMARKING OF ISWAP GATES… · pp. 15–16
- The iSWAP gate, defined as UEX(θ = π, ϕ = 0), is a Clifford gate and can therefore be characterized by randomized benchmarking.Source: RANDOMIZED BENCHMARKING OF ISWAP GATES… · p. 15
- For longer iSWAP gates, experimental error per gate agrees with simulation and indicates a limit set by the tunable coupler’s T ∗ 2 = 20 ns.At short gate lengths, error per gate increases due to leakage out of the computational subspace.Source: RANDOMIZED BENCHMARKING OF ISWAP GATES… · p. 16
HERENCE EFFECTS
The section evaluates how tunable-coupler coherence affects VQE accuracy by simulating the optimized ground-state energy at different coherence times. The simulations indicate that chemical accuracy requires T ∗ 2 greater than 500 ns.
HERENCE EFFECTS
- The section assesses the influence of tunable-coupler T ∗ 2 on VQE accuracy.Source: HERENCE EFFECTS · p. 17
- For each bond length, the simulation solves a Lindblad-type master equation and searches the θ, ϕ space for the global minimum energy.VQE accuracy is obtained by comparison with the energy from Hamiltonian diagonalization.Source: HERENCE EFFECTS · p. 17
- Chemical accuracy can be achieved in the simulations for tunable-coupler T ∗ 2 > 500 ns.Source: HERENCE EFFECTS · p. 17
REDUCED HAMILTONIAN FOR MOLECULAR HYDROGEN
The molecular-hydrogen calculation starts from a fermionic Hamiltonian in second quantization, with orbital creation and annihilation operators, and maps it to a qubit Hamiltonian using parity mapping. Its prefactors depend on one- and two-body interactions.
REDUCED HAMILTONIAN FOR MOLECULAR HYDROGEN
- For H2, the second-quantized Hamiltonian uses M = 4 molecular orbitals and includes one- and two-body interaction terms.Source: REDUCED HAMILTONIAN FOR MOLECULAR… · p. 18
- The Hamiltonian assumes spin conservation in the spin-orbital indices.Source: REDUCED HAMILTONIAN FOR MOLECULAR… · p. 19
- The molecular-hydrogen fermionic Hamiltonian is mapped to a qubit Hamiltonian using parity mapping.Source: REDUCED HAMILTONIAN FOR MOLECULAR… · p. 19
- The prefactors αi depend on the one- and two-body interaction terms and can be computed with PyQuante.Source: REDUCED HAMILTONIAN FOR MOLECULAR… · p. 19