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qubit-ADAPT-VQE: An adaptive algorithm for constructing hardware-efficient ansatze on a quantum processor

Ho Lun Tang, V. O. Shkolnikov, George S. Barron, Harper R. Grimsley, Nicholas J. Mayhall, Edwin Barnes, Sophia E. Economou

arXiv:1911.10205v2quant-ph

TL;DR

ADAPT-VQE lacked clear guidance on operator-pool selection, pool completeness, and practical circuit depth for near-term devices. This paper introduces qubit-ADAPT with a hardware-efficient Pauli-string pool, establishes a completeness criterion, and shows order-of-magnitude circuit-depth reductions with linear pool-size and measurement-overhead scaling.

  • Problem

    ADAPT-VQE’s operator-pool selection, required size, convergence guarantee, and practical circuit depth were not established for near-term quantum devices.

  • Method

    qubit-ADAPT uses Pauli strings as a hardware-efficient operator pool and defines completeness through operators capable of spanning the real n-qubit Hilbert space.

  • Results

    The method reduces circuit depth by an order of magnitude while maintaining the same accuracy as fermionic-ADAPT, and its minimal complete pool scales linearly with qubit number.

  • Takeaways & Limitations

    The linear minimal-pool scaling keeps qubit-ADAPT’s additional measurement overhead modest as system size increases.

Abstract

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Quantum simulation, one of the most promising applications of a quantum computer, is currently being explored intensely using the variational quantum eigensolver. The feasibility and performance of this algorithm depend critically on the form of the wavefunction ansatz. Recently in Nat. Commun. 10, 3007 (2019), an algorithm termed ADAPT-VQE was introduced to build system-adapted ansätze with substantially fewer variational parameters compared to other approaches. This algorithm relies heavily on a predefined operator pool with which it builds the ansatz. However, Nat. Commun. 10, 3007 (2019) did not provide a prescription for how to select the pool, how many operators it must contain, or whether the resulting ansatz will succeed in converging to the ground state. In addition, the pool used in that work leads to state preparation circuits that are too deep for a practical application on near-term devices. Here, we address all these key outstanding issues of the algorithm. We present a hardware-efficient variant of ADAPT-VQE that drastically reduces circuit depths using an operator pool that is guaranteed to contain the operators necessary to construct exact ansätze. Moreover, we show that the minimal pool size that achieves this scales linearly with the number of qubits. Through numerical simulations on $\text{H}_4$, LiH and $\text{H}_6$, we show that our algorithm ("qubit-ADAPT") reduces the circuit depth by an order of magnitude while maintaining the same accuracy as the original ADAPT-VQE. A central result of our approach is that the additional measurement overhead of qubit-ADAPT compared to fixed-ansatz variational algorithms scales only linearly with the number of qubits. Our work provides a crucial step forward in running algorithms on near-term quantum devices.

I. INTRODUCTION

VQE offers a near-term route to quantum simulation, but its success depends strongly on the chosen wavefunction ansatz. ADAPT-VQE reduces parameters through dynamic operator selection, while leaving unresolved questions about pool design, completeness, and circuit depth.

  • Motivation: VQE combines quantum energy measurements with classical parameter optimization and requires less coherence than quantum phase estimation.It has already been demonstrated on superconducting qubits, photons, and trapped ions.
  • Existing ansätze: UCCSD can require deep circuits, many parameters, and operator-ordering choices, while hardware-efficient ansätze may sacrifice systematic accuracy.These limitations motivate adaptive and hardware-aware ansatz construction.
  • ADAPT-VQE: ADAPT-VQE iteratively adds the pool operator producing the largest energy response, reducing parameters and improving accuracy relative to UCCSD in prior work.Its construction depends on a predefined operator pool.
  • Open problems: The original fermionic operator pool can produce impractically large circuits because fermion-to-spin mapping adds substantial gate overhead.The pool-selection rule, required pool size, and convergence guarantee were also left unclear.
  • Paper contribution: qubit-ADAPT addresses these issues with a hardware-efficient pool, reducing circuit depth by an order of magnitude while maintaining the same accuracy as fermionic-ADAPT.The paper also introduces a completeness criterion and proves that the minimal qualifying pool scales linearly with qubit number.
  • Paper contribution: The minimal pool result implies that qubit-ADAPT’s additional measurement overhead grows only linearly relative to fixed-ansatz VQE.The paper evaluates the approach through simulations of H4, LiH, and H6.

II. CIRCUIT DEPTH ESTIMATE FOR FERMIONIC-ADAPT

Fermionic-ADAPT selects operators by their energy gradients, but mapping fermionic excitations to Pauli strings creates high-CNOT state-preparation circuits. The estimated cost grows substantially with system size.

  • Adaptive construction: ADAPT grows the ansatz iteratively by selecting the pool operator with the largest energy response.The gradient is measured on the quantum device, and growth stops when its norm falls below a chosen threshold.
  • Measurement cost: ADAPT requires additional measurements roughly proportional to pool size times the number of Hamiltonian terms at each iteration.This overhead is incurred to evaluate the operator gradients.
  • Fermionic pool: Fermionic excitation pools are parameter-efficient, but Jordan-Wigner mapping makes them unlikely to be gate-efficient.Spin-adapted single and double excitations are used to construct the fermionic pool.
  • Scope: The Jordan-Wigner mapping used here is not generalized to other mappings, which the paper leaves for future work.This defines the scope of the circuit-depth analysis.
  • Mapping overhead: A Jordan-Wigner-mapped double excitation can contain up to 8 Pauli strings, producing high gate counts per excitation operator.The summed Pauli-string representation preserves spin and particle-number symmetries.
  • Circuit-depth estimate: 64m CNOTs is the large-m-orbital estimate for a spin-adapted doubles operator.The estimate follows first-order Trotterization and depends on the number of Pauli strings, Z operators, and summed spin-adapted doubles.

III. QUBIT-ADAPT

qubit-ADAPT replaces mapped fermionic excitation operators with selected Pauli-string operators to reduce gate costs. The resulting pool preserves relevant real-state structure while producing shallower circuits.

  • Qubit pool: Pauli strings are used as the qubit-ADAPT pool because each pool operator requires fewer CNOTs than a mapped spin-adapted fermionic excitation.This choice effectively reduces the average numbers of summed spin operators and Pauli strings to one.
  • Pool structure: The qubit pool contains odd Pauli strings, which are compatible with real fermionic operators and preserve a real ansatz under time-reversal symmetry.Even Pauli strings do not affect the energy because the relevant commutator expectation value vanishes for real states.
  • Numerical comparison: Across H4, LiH, and H6, qubit-ADAPT uses more parameters but substantially fewer CNOTs than fermionic-ADAPT.The comparison uses STO-3G calculations beginning from restricted Hartree-Fock orbitals without a frozen-core approximation.
  • Numerical comparison: The qubit pool reduces circuit depth by about an order of magnitude for H6 while maintaining comparable energy accuracy.The figure compares energy error against both ADAPT iterations and state-preparation CNOT counts.
  • Pool structure: The reduced qubit pool keeps operators appearing in the fermionic pool and uses strings with maximum length 4 when Z chains are omitted.This makes the pool smaller than all Pauli strings of maximum length 4 while retaining operators capable of transforming toward the ground state.

A. Numerical simulations

Numerical simulations compare qubit-ADAPT with fermionic variants and random operator orderings across H4, LiH, and H6, showing shallower circuits and faster convergence. Performance remains essentially unchanged across LiH bond distances, suggesting robustness to correlation strength.

  • The simulations use STO-3G restricted Hartree–Fock orbitals, with 8 spin-orbitals for H4 and 12 for LiH and H6.Bond distances are 1.5 Å for H4 and H6 and 2 Å for LiH, chosen where correlation effects are significant.
  • About an order of magnitude fewer CNOTs are used with the qubit pool for H6, although it requires more variational parameters than the fermionic pool.The added parameters shift computational cost toward classical optimization, while reduced CNOT counts ease quantum-processor demands.
  • 46–78 parameters are required for random H4 orderings to converge, compared with 30 for qubit-ADAPT.For LiH, random orderings require more than three times as many parameters as qubit-ADAPT, with only a lower bound available.
  • Qubit-ADAPT performance remains essentially the same across LiH bond distances from 1 Å to 3 Å.Overlapping curves indicate that the convergence rate changes little as the bond distance varies.

B. Operator pool reduction

The operator pool can be reduced substantially, but convergence requires completeness: the pool and its commutators must span enough operators to reach arbitrary real states. Minimal complete pools scale linearly with qubit number, while incomplete pools can stall despite vanishing gradients.

  • Pool reduction: 3/4 pool reduction leaves qubit-ADAPT performance similar to the original pool, whereas further removal can make the pool incomplete and prevent convergence.For a 1/32 pool, most H4 runs reach only an energy error of 10^-3.
  • Completeness: Completeness requires the pool operators and their commutators to transform the reference state into any real state in the n-qubit Hilbert space.The relevant operator set is generated through the Baker-Campbell-Hausdorff expansion.
  • Completeness: A pool is complete when its overlap matrix has rank r(M) ≥ 2^n − 1 for an arbitrary real state.The overlap matrix is defined by M_ij = ⟨ψ|A_i†A_j|ψ⟩.
  • Minimal pools: 2^n − 2 operators suffice for minimal complete pools, far fewer than the exponentially sized Hilbert space and the fermionic pool scaling as n^4.Numerical tests found complete pools at this size, and constructive families establish existence for any n.
  • Completeness: 20–40% of randomly selected pools containing 2^n − 2 operators are complete, while complete pools converge and incomplete pools may fail to reach the ground state.The comparison used random real Hamiltonians with 3, 4, and 5 qubits and minimal complete pools {V_j}_n and {G_j}_n.
  • Measurement overhead: Because the minimal pool size is linear in n, qubit-ADAPT’s additional measurement overhead per iteration also remains linear in n.This addresses the larger qubit-pool measurement cost relative to the fermionic pool.

IV. CONCLUSIONS

The paper introduces qubit-ADAPT, a hardware-efficient ADAPT-VQE variant using Pauli strings, and establishes completeness conditions and constructive minimal pools. These results reduce state-preparation depth and measurement requirements for near-term hardware.

  • Method: Qubit-ADAPT replaces fermionic operators with Pauli strings, reducing the CNOT cost associated with each pool operator.The method is designed as a more efficient, NISQ-compatible ADAPT-VQE variant.
  • Guarantee: A completeness condition guarantees that a pool can generate an exact ADAPT ansatz.The condition is tied to the operators and their commutators spanning the required state space.
  • Pool size: The smallest pool satisfying the completeness criterion scales linearly with the number of qubits.The paper also gives a constructive procedure for generating minimal complete pools for arbitrary qubit counts.
  • Consequences: The resulting approach substantially reduces state-preparation circuit depth and the number of measurements needed for realistic hardware.These are the paper’s stated practical consequences of the qubit-ADAPT construction.

Appendix A: CNOT estimation

The paper estimates CNOT requirements from the structure of pool operators and uses a conservative assumption that selected operators are four-qubit Pauli strings.

  • CNOT estimation: CNOT count is estimated as a function of the number of variational parameters from the structure of the pool operators.The estimate assumes every selected operator is a four-qubit Pauli string for conservativeness.

Fermionic ADAPT

Fermionic-ADAPT maps spin-adapted fermionic operators to Pauli strings, enabling circuit-depth estimates from operator, Pauli, and spin-adaptation counts. The estimate is accurate for H4 and H6 when spin orbitals are evenly explored, but assumes no core orbitals.

  • Operator grouping: Spin-orbital combinations are classified into five groups because groups differ in whether they contain singlet and triplet operators and in their fermionic-term counts.The calculation also tracks the resulting Pauli-string counts and Pauli-Z-chain lengths.
  • Jordan-Wigner mapping: The Jordan-Wigner transformation converts fermionic anti-Hermitian operators into Pauli strings, with string counts depending on index coincidences and operator groups.Pairs with four distinct indices produce eight Pauli strings, while repeated-index cases can produce fewer.
  • Numerical estimate: For H4, the estimated CNOT count is 1928.41, using 11 parameters and the averaged Pauli, Pauli-Z, and spin-adaptation factors.The estimate is reported as close to the number obtained directly.
  • Circuit-depth estimation: Fermionic-ADAPT estimates CNOT requirements by combining the number of ansatz parameters, Pauli strings, Pauli-Z chains, and spin-adapted fermionic terms.The stated estimate is NCNOT ≈ Npars × NPauli × (6 + 2 × NZ) × Nspin.
  • Scope: The estimation is accurate for H4 and H6 when all spin orbitals are evenly explored, meaning the systems contain no core orbitals.This condition is stated as the scope of the estimation's accuracy.

Appendix B: Constructive proof of minimal complete pools

The appendix proves constructively that complete qubit-ADAPT pools exist with only 2n−2 operators. These pools can rotate any real state into any other real state.

  • Constructive result: Complete qubit-ADAPT pools containing only 2n−2 operators are called minimal complete pools.The proof begins with an explicit complete pool for three qubits and extends it recursively.
  • State reachability: Using operators generated by the pool, any real state can be rotated into any other real state in the Hilbert space.The construction uses products of exponentials of pool operators.

1. Complete pool for 3 qubits

For three qubits, the pool {iZ3Z2Y1, iZ3Y2, iY3, iY2} is shown to be minimal and complete by mapping arbitrary states to |000⟩.

  • Complete pool for 3 qubits: The three-qubit pool is {iZ3Z2Y1, iZ3Y2, iY3, iY2}.Completeness is established by constructing a transformation from an arbitrary state to |000⟩.
  • Norm equalization: A Y3 rotation equalizes the norms of the two conditional two-qubit states in an arbitrary three-qubit state.The rotation angle is chosen so the resulting conditional states have equal norms.
  • State factorization: Conditional rotations generated by the reduced pool factor out qubit 3 while preserving the required state structure.The reduced pool omits iY3 and contains {iZ3Z2Y1, iZ3Y2, iY2}.
  • Completion: A final Y3 rotation followed by a conditional qubit-1 operation maps the factored state to |000⟩.This completes the constructive transformation for arbitrary three-qubit states.
  • Completeness: Because every three-qubit state can be mapped to |000⟩, the same pool can map any three-qubit state to any other three-qubit state.The pool is therefore complete and minimal for three qubits.

2. Proof of the complete pool for n qubits

The three-qubit construction is extended inductively to n qubits using recursively defined pools. The proof shows these pools are complete and contain 2n−2 operators.

  • Inductive step: Conditional rotations and the induction hypothesis factor the left-most qubit while preserving the required structure of the remaining terms.Additional conditional operations handle norm equalization and rotations of the remaining qubits.
  • Inductive theorem: The proof establishes a theorem for factoring one qubit from a generic n-qubit state using a reduced n-qubit pool.The theorem is proved by induction, beginning with the three-qubit construction.
  • Extension: The proof extends to factoring the (n+1)th qubit by using operators that reproduce the necessary conditional rotations and composite unconditional rotations.The constructed operators preserve the relevant qubit states before restoring them after the rotation.
  • Induction conclusion: The induction is completed for n+1 qubits, showing that the factorization procedure works beyond the three-qubit base case.The result follows after applying the induction hypothesis and the required conditional rotations.
  • Complete pool: The recursively defined pool is complete for n+1 qubits and contains 2n−2 operators.It can rotate an arbitrary state to |0⟩⊗n+1 and therefore between any two states.

Appendix C: Mapping between minimal complete pools

Appendix C establishes that the iteratively constructed pool {V_i} and the adjacent-qubit pool {G_i} share an operator basis through commutator relations. Because {V_i} is complete, this mapping proves that {G_i} is also a minimal complete pool for any number of qubits.

  • Pool construction: The appendix compares two minimal complete pools: iteratively constructed operators {V_i} and operators acting only on adjacent qubits in a linear array, {G_i}.The elements are ordered explicitly for the subsequent mapping analysis.
  • Pool construction: The Pauli-string operators in {V_i} are listed as strings containing Y, Z, and identity operators across the qubits.The sequence includes V_1 through V_2n−2 with the displayed string patterns.
  • Pool construction: The adjacent-qubit operators {G_i} are likewise listed as Pauli strings, with Y and Z positioned along the linear qubit array.The sequence runs from G_1 through G_2n−2.
  • Commutator mapping: The V_i operators can be obtained from products and commutators of the G_i operators, so the two pools share the same operator basis.The appendix gives explicit product relations, including V_n = G_n and relations for V_2n−k.
  • Completeness: Since the V_i pool is complete, the shared basis proves that the adjacent-qubit G_i pool is minimal and complete for any number n of qubits.The product operators must be ordered with smaller index values to the left and larger values to the right.
  • Commutator mapping: Two Pauli strings fail to commute when they differ by an odd number of non-identity Pauli operators, and their commutator is proportional to their product.This rule supplies the algebraic step used to relate the two pools.
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