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Efficient quantum circuits for quantum computational chemistry

Yordan S. Yordanov, David R. M. Arvidsson-Shukur, Crispin H. W. Barnes

arXiv:2005.14475v2quant-ph

TL;DR

Efficient circuits for fermionic excitations are a practical challenge for VQE ansatz states on NISQ computers. This paper builds qubit-excitation circuits and extends them to account for fermionic anticommutation, reducing CNOT counts by factors of 2 and 8 for single and double excitations compared with CNOT-staircase circuits.

  • Problem

    VQE requires ansatz states built from fermionic excitations, but efficient circuits for creating these states are challenging and entangling-gate fidelity is a NISQ bottleneck.

    Source: §I · p. 1

  • Method

    The authors construct qubit-excitation circuits and modify them to account for fermionic anticommutation relations.

    Source: §I · p. 1

  • Results

    CNOT counts are reduced by factors of 2 and 8 per single and double fermionic excitation, respectively, versus circuits constructed entirely with CNOT staircases.

    Source: §I · p. 1

  • Takeaways & Limitations

    The optimized circuits are a significant step toward implementing VQE algorithms on NISQ devices.

    Source: §IV · p. 6

Abstract

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Molecular quantum simulations with the variational quantum eigensolver (VQE) rely on ansatz states that approximate the molecular ground states. These ansatz states are generally defined by parametrized fermionic excitation operators and an initial reference state. Efficient ways to perform fermionic excitations are vital for the realization of the VQE on noisy intermediate-scale quantum computers. Here, we address this issue by first demonstrating circuits that perform qubit excitations, excitations that do not account for fermionic anticommutation relations. We then extend the functionality of these circuits to perform fermionic excitations. Compared to circuits constructed with the standard use of "$CNOT$ staircases", our circuits offer a linear reduction in the number of $CNOT$ gates, by a factor of $2$ and $8$ per single and double excitation, respectively. Our results reduce the requirements for near-term realizations of quantum molecular simulations.

I. INTRODUCTION

The introduction identifies efficient ansatz-state circuits as a practical VQE challenge, then presents a staged construction of qubit and fermionic excitation circuits. The theory section sets the Jordan–Wigner encoding and reviews fermionic excitations and their standard CNOT-staircase implementation.

I. INTRODUCTION

  • VQE ansatz design and efficient circuits to create ansatz states are practical challenges, while high-fidelity entangling gates are a NISQ bottleneck.The paper motivates more efficient fermionic-excitation circuits against this hardware constraint.Source: §I · p. 1
  • The proposed construction first implements qubit excitations, then accounts for fermionic anticommutation, reducing CNOT counts by factors of 2 and 8 for single and double excitations versus CNOT-staircase circuits.The method also optimizes CNOT circuit depth.Source: §I · p. 1

II. THEORY

  • The paper assumes Jordan–Wigner encoding, where each molecular spin-orbital occupancy is represented by a qubit state.Spin-orbital indices are ordered i < j < k < l.Source: §II · p. 1

A. Fermionic excitations

  • A fermionic excitation is an exponential of parametrized fermionic excitation operators, whose creation and annihilation operators obey anticommutation relations.The paper reviews standard circuits built from CNOT staircases; its single-excitation construction uses two such constructions, while the double-excitation construction uses eight.Source: A. Fermionic excitations · p. 2

B. Multi-qubit-controlled rotations

The paper describes an ancilla-free method for implementing multi-qubit-controlled rotations. It decomposes a controlled rotation into opposite half-angle rotations and combines these decompositions to cancel adjacent CNOT gates.

B. Multi-qubit-controlled rotations

  • The controlled-rotation method uses no ancilla qubits and is later used to construct qubit-excitation circuits.The target rotation is conditioned on the controls being in the all-ones state.Source: B. Multi-qubit-controlled rotations · p. 3
  • An m-qubit-controlled rotation is decomposed into two opposite half-angle rotations, each controlled by m − 1 qubits.The controlled rotation acts on target q0, with controls {q1..qm}.Source: B. Multi-qubit-controlled rotations · p. 3
  • For m > 2, alternating Eqs. (12) and (13) cancels adjacent CNOTs and yields a circuit with 2^(m−1) + 2 CNOTs.In the illustrated circuit, the two middle CNOTs between q0 and q1 can be cancelled.Source: B. Multi-qubit-controlled rotations · p. 3

III. RESULTS

The results section introduces qubit excitations and circuits for single and double qubit excitations, then adapts those circuits to perform fermionic excitations.

III. RESULTS

  • The section introduces qubit excitations before presenting circuits for single and double qubit excitations.The supplied section text does not report their results.Source: §III · p. 3
  • Qubit excitations are the initial circuit target in the section’s progression toward fermionic excitations.This point summarizes the stated order of the results section.Source: §III · p. 3
  • The qubit-excitation circuits are modified to perform single and double fermionic excitations instead.Source: §III · p. 3

A. Qubit excitations

Qubit excitations are generated by qubit creation and annihilation operators whose relations are neither bosonic nor fermionic. Unlike fermionic excitations, they omit the Pauli-z products that account for fermionic anticommutation relations.

A. Qubit excitations

  • Qubit excitations are unitary evolutions generated by exponentials of parametrized qubit-excitation operators built from Q and Q†.The single- and double-excitation operators are skew-Hermitian.Source: A. Qubit excitations · p. 3
  • The operators Q and Q† satisfy relations that are neither bosonic nor fermionic; some authors call them parafermionic.Source: A. Qubit excitations · p. 3
  • Unlike fermionic excitations, qubit excitations lack the Pauli-z products that account for fermionic anticommutation relations.These operations have also been considered in VQE-ansatz construction.Source: A. Qubit excitations · p. 3

1. Circuit for single qubit excitations

The single qubit-excitation unitary is equivalent to an exchange interaction and is implemented with an explicit circuit. The circuit shown is chosen for its role in constructing later circuits, though it is not gate-count optimal on its own.

1. Circuit for single qubit excitations

  • The single qubit-excitation unitary is equivalent to an exchange-interaction evolution, implemented by the circuit in Fig. 3.Source: §1 · p. 3, Figure 3 · p. 4
  • A single qubit excitation can use two CNOTs, so Fig. 3b is not gate-count optimal by itself; it supports optimal constructions in Figs. 5, 8, and 9.Source: §1 · p. 3
  • Fig. 3b implements the exchange circuit using a controlled-Ry rotation and the Fig. 4 identity to cancel a CNOT.Source: §1 · p. 3, Figure 3 · p. 4, Figure 4 · p. 4

2. Circuit for double qubit excitations

The double qubit-excitation circuit exchanges a specific pair of states and leaves other states unchanged, using parity controls to restrict the operation. Its explicit implementation matches the cited circuits' CNOT count while reducing depth.

2. Circuit for double qubit excitations

  • The operation continuously exchanges |1i1j0k0l⟩ and |0i0j1k1l⟩ while leaving other states unchanged.Source: §2 · p. 4
  • CNOTs encode the parities of pairs {qk, ql} and {qi, qj} on qk and qi, which control exchange between ql and qj before the parity encoding is reversed.Source: §2 · p. 4
  • 13 CNOTs and depth 11 implement the double qubit excitation, matching Refs. [32] in count but improving on their depth of 13.Fig. 7 shows the explicit circuit.Source: §2 · p. 4, Figure 7 · p. 5

B. Efficient Fermionic excitations

The fermionic circuits add parity-dependent sign changes to the qubit-excitation circuits. This construction uses fewer CNOT staircases than the standard approach for both single and double excitations.

B. Efficient Fermionic excitations

1. Circuit for single fermionic excitations

The modified single-fermionic-excitation circuit encodes intermediate-qubit parity and conditionally reverses a controlled rotation, reducing CNOT use relative to the standard construction.

1. Circuit for single fermionic excitations

  • The CNOT staircase encodes the parity of intermediate qubits; conditioned on qi−1 being |1⟩, two CZ gates reverse the Ry(θ) rotation to Ry(−θ).A circuit identity cancels a CNOT between qk and qi.Source: §1 · p. 5
  • For n(sf) ≥ 3, the circuit uses 2n(sf) − 1 CNOTs with depth max[5, 2n(sf) − 3], versus 4(n(sf) − 1) CNOTs and depth in the standard construction.For n(sf) = 2, it is equivalent to the single-qubit-excitation circuit.Source: Figure 8 · p. 5, §1 · p. 5
  • The circuit acts on qubits qi through qk, with the figure's vertical dots marking staircases across qi+1 to qk−1.Source: Figure 8 · p. 5, §1 · p. 5

2. Circuit for double fermionic excitations

The double-fermionic-excitation circuit uses a parity staircase and controlled-rotation reversal, with CNOT count and depth formulas reported alongside a balanced-tree option for reducing depth growth.

2. Circuit for double fermionic excitations

  • The staircase encodes the parity of qubits qi+1 through qj−1 and qk+1 through ql−1; two CZ gates conditionally reverse the controlled Ry rotation.A circuit identity cancels a CNOT between ql and qj.Source: §2 · p. 5
  • The double-fermionic circuit uses 2n(df) + 5 CNOTs and depth max[13, 2n(df) − 1] for n(df) ≥ 5; the authors report improved count and depth over previous double-qubit and double-fermionic methods.The figure depicts the double-fermionic-excitation circuit.Source: §2 · p. 5, Figure 9 · p. 6, §IV · p. 6
  • The standard fermionic-excitation construction uses 16(n(df) − 1) CNOTs and has depth 16(n(df) − 1), compared with the proposed circuit's 2n(df) + 5 CNOTs and max[13, 2n(df) − 1] depth.Source: §2 · pp. 5–6
  • Balanced-tree rearrangement makes the fermionic-excitation circuits' CNOT depth grow logarithmically rather than linearly.Source: §2 · p. 6

IV. SUMMARY

  • Single-fermionic-excitation circuits use 2n(sf) − 1 CNOTs, with depth 3 for n(sf) = 2 and max[5, 2n(sf) − 3] for n(sf) ≥ 3.Here n(sf) is the number of orbitals on which the excitation acts.Source: §IV · p. 6

Appendix A: Standard single and double fermionic excitations circuits

The appendix presents the standard CNOT-staircase circuits for single and double fermionic excitations.

Appendix A: Standard single and double fermionic excitations circuits

  • The appendix supplies the full standard circuits for both single and double fermionic excitations.Source: Appendix A · p. 6
  • The standard single-fermionic-excitation circuit contains two CNOT staircase constructions.The figure shows the circuit for Eq. (10).Source: Figure 10 · p. 6, Appendix A · p. 6
  • The standard double-fermionic-excitation circuit contains eight CNOT staircase constructions.The figure shows the circuit for Eq. (11).Source: Appendix A · p. 6, Figure 11 · p. 7
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