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Efficient Symmetry-Preserving State Preparation Circuits for the Variational Quantum Eigensolver Algorithm
Bryan T. Gard, Linghua Zhu, George S. Barron, Nicholas J. Mayhall, Sophia E. Economou, Edwin Barnes
TL;DR
VQE needs trial-state ansätze that contain relevant energy eigenstates while avoiding low-overlap or unphysical sectors on NISQ hardware. This paper develops symmetry-preserving preparation circuits with minimal parameters for the appropriate subspaces and tests them on molecular simulations. The circuits are reported to improve accuracy and circuit depth relative to standard state-preparation methods.
Problem
VQE efficiency depends on preparing trial states that include relevant energy eigenstates while avoiding states with little overlap with them.
Method
The paper constructs state-preparation circuits preserving particle number, total spin, spin projection, and time-reversal with the minimal parameters needed for each symmetry subspace.
Results
The circuits are reported to outperform standard state-preparation methods in both accuracy and circuit depth in H2 and LiH simulations.
Takeaways & Limitations
Symmetry-preserving ansätze restrict VQE exploration to relevant Hilbert-space sectors while retaining the ground state within the spanned space.
Takeaways & Limitations
The number of variational parameters still grows exponentially with system size, making complete symmetry-subspace spanning impractical beyond a few tens of qubits.
Abstract
from arXiv · showhide
The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the $H_2$ and $LiH$ molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.
I. INTRODUCTION
VQE uses variational ansätze to prepare trial states on shallower NISQ-compatible circuits, but ansatz design must balance hardware feasibility with coverage of physically relevant Hilbert-space sectors. The paper motivates symmetry-preserving circuits as a systematic way to span desired subspaces while excluding unphysical states.
- VQE prepares and measures multi-qubit states with a variational ansatz whose parameters are optimized classically.
- Ansatz design is crucial because it can determine VQE success on NISQ devices.
- Hardware-based ansätze are NISQ-friendly but can be ad hoc and cause barren plateaus as qubit number and Hilbert-space dimension increase.
- Hardware-based ansätze must span the Hilbert-space region containing the solution while avoiding unphysical states.
- The paper introduces circuits preserving particle number, total spin, spin projection, and time-reversal while using the minimal parameters needed to span the relevant symmetry subspace.
- The circuits are designed for arbitrary orbital, electron, and spin quantum-number counts, with hardware constraints including reduced CNOT requirements.
A. Particle number and time-reversal symmetries
The paper constructs particle-number-preserving circuits that span the relevant electron subspace with the minimal number of real parameters, and shows how to impose time-reversal symmetry by restricting phases. These circuits can be systematically extended to arbitrary orbital and particle numbers, with trade-offs between generality, connectivity, and CNOT count.
- Particle-number symmetry: Fixing m electrons among n spin-orbitals defines a subspace of dimension dim(Hn,m), requiring 2 dim(Hn,m) − 2 real parameters for arbitrary complex states.Normalization and global phase remove two real degrees of freedom.
- Particle-number symmetry: The two-qubit one-excitation circuit spans α |01⟩ + β |10⟩ exactly with two parameters, saturating the lower bound for H2,1.The circuit uses an initial X gate followed by the parameterized A gate.
- Particle-number symmetry: The exchange-type A gate preserves particle number by mixing |01⟩ and |10⟩ while leaving |00⟩ and |11⟩ unchanged.A preceding X gate can place the state in the desired excitation sector before A gates generate superpositions within it.
- Gate decomposition: The A gate decomposes into two single-qubit rotations and three CNOT gates, with the three-CNOT decomposition minimal in CNOT count.The rotations are R(θ, φ) = Rz(φ + π)Ry(θ + π/2).
- Time-reversal symmetry: For n = 4 and m = 2, the construction spans six basis states with the minimal 10 parameters, while time reversal reduces the count to 6.Time reversal is imposed by setting all φi parameters to zero and fixing one additional θ parameter.
- General construction and trade-offs: The general construction uses X gates and a cascade of A gates, reaches unit fidelity at the minimal parameter count, and requires only nearest-neighbor coupling.A case-specific alternative uses 9 CNOT gates for n = 4, m = 2, versus 18 for the general construction, but requires non-neighboring couplings.
B. Spin Symmetries
The paper develops circuits that preserve particle number, spin projection, and total spin while spanning the corresponding symmetry subspaces. These reductions can substantially shrink the relevant Hilbert space, although the scaling remains exponential, and simulations expose a trade-off between parameter minimality and CNOT cost under noise.
- Motivation: The authors identify simultaneous conservation of particle number, spin projection, and total spin as an open state-preparation problem and introduce a construction addressing it.The approach avoids imposing nonlinear parameter constraints on particle-number circuits.
- Circuit construction: The spin-symmetry construction separates spin-up and spin-down orbitals and sets parameters on gates bridging the two spin subspaces to zero.This prevents mixing between the two spin subspaces while retaining the broader cascade construction.
- Particle-number representation: For n spin-orbitals, fixing electron number corresponds to fixing the number of qubits in state |1⟩ under the Jordan-Wigner mapping.This makes particle-number sectors directly representable as qubit-excitation subspaces.
- Hilbert-space reduction: For n = 28 spin-orbitals, imposing all symmetries makes the relevant subspace at least two orders of magnitude smaller than the full Hilbert space, while scaling remains exponential.The reduction can lower quantum-processor demands and accelerate classical VQE optimization.
- Circuit construction: The general procedure constructs circuits for arbitrary valid n, m, s, and sz, using recursive hyperspherical coordinates to eliminate basis states and reduce the effective Hilbert-space dimension.The resulting circuits span the target symmetry subspace by construction.
- Simulation results: In noisy simulations, A-gate circuits provide a middle ground between parameter count and CNOT count, whereas exactly parameter-minimal E-gate circuits suffer from larger CNOT counts.For LiH mapped to 6 qubits, the E-gate ansatz shows notably more error; the authors therefore consider relaxing some symmetry constraints favorable under noise.
III. DISCUSSION
The paper presents symmetry-preserving state-preparation schemes for quantum simulation, covering particle number, time-reversal, total spin, and spin magnetization. These ansätze contain the ground state with the minimal parameters set by the symmetry subspace, but their parameter count still grows exponentially with system size.
- III. DISCUSSION: The proposed circuits systematically preserve particle number, time-reversal, total spin, and spin magnetization.The paper provides general construction procedures, explicit circuit examples, and gate counts.
- III. DISCUSSION: Enforcing spin symmetries can improve VQE by preventing exploration of vast, irrelevant regions of Hilbert space.
- III. DISCUSSION: The ansätze are guaranteed to contain the ground state while using the minimal number of parameters defined by the symmetry subspace.
- III. DISCUSSION: The number of variational parameters still grows exponentially with system size, making full symmetry-subspace spanning impractical beyond a few tens of qubits.
IV. METHODS
The study validates circuit compilation and state-space coverage numerically, then implements the proposed symmetry-enforcing ansätze in Qiskit for noisy simulations using IBM hardware noise parameters.
- IV. METHODS: Mathematica and numerical optimization were used to confirm that the proposed circuits span arbitrary states in the desired Hilbert space.Simulated annealing was identified as the best-performing NMaximize method across many random states.
- IV. METHODS: The VQE simulations were implemented in IBM's Qiskit software using the proposed ansätze with relevant symmetries enforced.
- IV. METHODS: Noisy simulations used the noise parameters of IBM's Poughkeepsie device.
A. Construction of E gate circuits
The E-gate construction builds unitaries that prepare arbitrary states within selected particle-number and spin subspaces, then decomposes them into standard gates or numerical circuits.
- A. Construction of E gate circuits: The construction starts with unitaries that transform the all-zero state into arbitrary states in a chosen particle-number subspace.The target coefficients are specified using hyperspherical coordinates.
- A. Construction of E gate circuits: For two fermions in four spin-orbitals, an explicit unitary is constructed with a first column matching the target state and remaining columns chosen for unitarity.
- A. Construction of E gate circuits: The symbolic decomposition uses 4-qubit Toffoli gates, with a single such gate decomposing into 13 CNOT gates.
- A. Construction of E gate circuits: Numerical decompositions use quantum multiplexers to prepare the same output state as the proposed E gate while reducing CNOT requirements.The method is distinct from decomposing the proposed E4 gate itself.
- A. Construction of E gate circuits: For three fermions in six spin-orbitals, an analogous E6 unitary is decomposed into Toffoli gates and compared with numerical state-preparation methods.The numerical method constructs the desired superposition in 124 CNOT gates.
B. Numerical Verification of Circuits
Numerical tests verify that the proposed circuits reach arbitrary states in their target symmetry subspaces with the minimal parameter count. Spin restrictions reduce the required parameters, while unconstrained ansätze can encounter difficult near-degenerate optimization landscapes.
- B. Numerical Verification of Circuits: The number-conserving n = 4, m = 2 circuit spans a six-basis-state subspace using the fewest possible parameters under the tested constraints.
- B. Numerical Verification of Circuits: Fidelity increases monotonically with the number of parameters and reaches unity when that number equals the symmetry-subspace dimension.This behavior was observed across all tested n and m combinations.
- B. Numerical Verification of Circuits: For the sz = 0 restriction, modifying the base circuit preserves the symmetry without mixing the split spin subspaces.
- B. Numerical Verification of Circuits: Three parameters suffice to span the relevant restricted subspace and achieve unit fidelity in the tested case.