A comparison of the Bravyi-Kitaev and Jordan-Wigner transformations for the quantum simulation of quantum chemistry

1 Introduction

• Full configuration interaction scales factorially with the number of basis functions, limiting its application to small systems.The paper motivates quantum simulation as a possible way to address this computational difficulty.Source: §1 · pp. 2–3
• The canonical electronic-structure simulation pipeline maps molecular orbitals and the Hamiltonian to qubits, forms an evolution operator, then estimates ground-state energy.The described steps use Trotter-Suzuki approximation and phase estimation.Source: §1 · pp. 3–4
• The comparison addresses a gap: numerical assessments of Bravyi-Kitaev performance had not examined many systems at large scale.Earlier examples included hydrogen and methane, with relatively modest overall gate savings reported for methane despite entangling-gate savings.Source: §1 · p. 4
• Across 86 molecular systems, Bravyi-Kitaev circuits are typically equal in length to or shorter than Jordan-Wigner circuits.The paper also reports substantially reduced gate-count estimates for Bravyi-Kitaev with limited circuit optimisation.Source: §1 · p. 4
• The study also examines how Trotter ordering affects gate counts and Trotter error across a subset of systems.Ordering can increase total gate counts through its effect on approximation error, even when it reduces gates within each Trotter step.Source: §1 · pp. 4–5

2 Theory

• The classical difficulty arises from the Fock-space dimension, which scales exponentially with basis orbitals and remains intractable even when fixed-electron restriction reduces scaling to factorial.The number of integrals scales quartically, but they are classically computable and molecular symmetries often reduce the nonzero count.Source: §2 · p. 5
• Trotter–Suzuki approximation produces a circuit for molecular-Hamiltonian evolution and adds simulation error, which can be reduced by increasing steps at extra multiplicative computational expense.The broader quantum workflow prepares an approximate ground state and uses phase estimation to determine its energy.Source: §2 · p. 6

2.1 The Bravyi-Kitaev Mapping

• Bravyi–Kitaev combines occupation and parity information and simulates a fermionic operation in O(log2 n) qubit operations, compared with Jordan–Wigner’s O(N) scaling.Both mappings use N qubits; the paper examines their resource implications.Source: §2.1 · pp. 8–9, §2 · p. 8
• At most 4 log2 i+2 gates are required to simulate one fermionic operation, fewer than Jordan–Wigner’s i gates when i ≥19; Bravyi–Kitaev is conservatively expected to reduce cost above N ≈32 spin-orbitals.Source: §2.1 · p. 10

2.2 Trotterization and Simulation

• The Trotterization subsection begins by identifying simulation of the unitary evolution operator as the circuit task needed for phase estimation of molecular ground-state energy.Source: §2.2 · p. 11
• The Hamiltonian is a weighted sum of Pauli-operation strings, whose exponentials are sequenced using a first-order Trotter-Suzuki approximation over n steps.The overall evolution time is subdivided into n steps.Source: §2.2 · p. 11
• Standard techniques convert the exponentiated Pauli strings into circuits with single-qubit rotations and typically more expensive two-qubit entangling gates, implemented sequentially.Source: §2.2 · pp. 11–12
• Increasing the number of Trotter steps reduces the approximation error introduced by Trotter-Suzuki simulation.The paper examines the impact of this error in section 5.Source: §2.2 · p. 12

3 Basic Circuits

• The study assessed serial quantum circuit lengths for 86 small molecules and atoms optimized at the Hartree–Fock level, spanning 2–54 spin-orbitals; most used STO-3G.Larger Pople basis sets were used in 18 trials. The authors state that the chosen basis is insufficient for an accurate electronic-structure solution.Source: §3 · p. 12
• The workflow generated qubit Hamiltonians as Pauli strings, then optimized circuits at the qubit level, allowing greater flexibility in ordering terms for Trotterization.Hartree-Fock integrals were obtained using PSI4 and its FermiLib plugin; the code did not retain the original fermionic components during circuit optimization.Source: §3 · pp. 13–14
• Preliminary tests with natural and orthogonalised atomic orbital integrals found no consistent dependence of relative mapping performance on basis choice; rigorous confirmation remains future work.The basis-choice comparison was omitted from the main simulations to reduce computational cost.Source: §3 · pp. 13–14
• The analysis counted gates for one Trotter step, assumed larger step counts scale as multiples of that count, and omitted a full phase-estimation and error-correction resource analysis.It also neglected possible gate savings at interfaces between Trotter steps.Source: §3 · p. 14
• The mappings require the same number and type of non-Clifford gates, so choosing between them does not change circuit T count or error-correction difficulty.The authors attribute this to the Hamiltonians having the same number of terms under either mapping.Source: §3 · p. 15
• From around 30 spin-orbitals, Bravyi-Kitaev reliably produced shorter unoptimised circuits, with up to around 25% gate savings in examples with 50 spin-orbitals.Figure 3 also shows Jordan-Wigner outperforming Bravyi-Kitaev in some instances.Source: Figure 3 · p. 16
• Bravyi–Kitaev shows consistent improvement from roughly 30 spin-orbitals, amounting to approximately 25% of overall circuit length for the largest examined systems; smaller systems may show no improvement or longer circuits.The crossover where this advantage dominates increased overhead is around 32 spin-orbitals.Source: §3 · p. 17
• Cancellation saves CNOTs by retaining parity information and basis changes across Hamiltonian terms instead of recomputing parities or resetting bases unnecessarily.Much of the cancellation savings comes from maintaining the calculation basis between sequential terms rather than reducing entangling gates.Source: §3 · pp. 18, 20
• After duplicate gates are removed, circuits involving more than about 107 gates use Bravyi-Kitaev mapping to produce circuits approximately 30-40% shorter than Jordan-Wigner circuits with magnitude ordering.Source: §3 · p. 18, Figure 4 · p. 19
• Under magnitude ordering, no optimized Jordan-Wigner circuit outperformed Bravyi-Kitaev for systems requiring more than 10^6 gates, leading the authors to favor Bravyi-Kitaev generally.The comparison uses magnitude ordering for both mappings.Source: §3 · pp. 19–20
• Bravyi-Kitaev's general gate savings include substantially fewer CNOT gates, typically offset by a small increase in single-qubit gates.Gate cancellation itself yields relatively few entangling-gate savings compared with the mapping advantage.Source: §3 · pp. 19–20
• Further CNOT optimization is difficult without reducing circuit locality, and changing Trotter ordering may increase error and require more steps, potentially undermining ordering-related savings.The reported results use magnitude ordering; the effect of Trotter ordering is considered separately.Source: §3 · p. 20

• With lexicographic ordering, Jordan-Wigner circuits are marginally shorter overall and about 25% shorter for the longest circuit, propanol, because Bravyi-Kitaev's less-linear CNOT chains hamper cancellation.The savings from lexicographic ordering outweigh Bravyi-Kitaev's savings under magnitude ordering in longer circuits.Source: Figure 5 · p. 22, §4 · pp. 22, 25
• The comparison uses randomized, lexicographic, magnitude, and interleaved term orderings; exhaustive ordering search is intractable for non-trivial systems because possibilities grow factorially.Source: §4 · pp. 22–23
• For every non-lexicographic ordering, Bravyi-Kitaev outperforms Jordan-Wigner; beyond N = 10, its advantage exceeds gate cancellation alone, reaching about an order of magnitude for CNOT savings with magnitude ordering.The reported comparison concerns gate counts; it does not include Trotter-error implications.Source: §4 · p. 23
• For the largest example, random ordering yields a Bravyi-Kitaev CNOT-count reduction forty times that from gate cancellation alone, although one random ordering is not statistically representative.Its qualitative trend largely matches the other ordering schemes.Source: §4 · p. 24
• Choosing an ordering to minimize Trotter error may require one that is suboptimal for single-step gate count, even when Bravyi-Kitaev has a gate-count advantage.Source: §4 · p. 25
• Gate-cancellation-optimized ordering does not predictably improve Bravyi-Kitaev gate counts, unlike the substantial, predictable gains found with other orderings, particularly random ordering.Source: §4 · p. 25

5 Trotter Error Considerations

• Error analysis covered 34 systems because exact ground-state vectors remain exponentially difficult to obtain, with examples above 20 spin-orbitals requiring hundreds of gigabytes of RAM.Determining exact Trotter error is itself exponentially hard.Source: §5 · p. 26
• The authors suggest fewer than 10 Trotter steps may suffice for the first precision bit in larger systems, while error compounds at higher phase-estimation precision.This projection follows from small-system errors and is not a full phase-estimation error analysis.Source: §5 · p. 27
• One first-order Trotter step yielded chemical accuracy for a single evolution-operator application, while error was often below 0.001 Hartree in the overwhelming majority of systems.The authors note that higher powers would need assessment for full phase estimation.Source: §5 · p. 27, Figure 7 · p. 28
• Magnitude ordering usually lowered Trotter error relative to lexicographic ordering, sometimes by over an order of magnitude, but this was variance on exceptionally small errors.The authors do not claim this constitutes a useful error reduction because one step already sufficed for chemical accuracy in most studied systems.Source: §5 · p. 27, Figure 7 · p. 28
• For magnitude ordering, Jordan-Wigner and Bravyi-Kitaev errors were nearly identical in almost all cases; under lexicographic ordering, Bravyi-Kitaev had lower error in most systems.In some lexicographically ordered systems, the Bravyi-Kitaev error was almost halved.Source: Figure 8 · p. 29, §5 · p. 29

6 Further Optimized Circuits

• The circuit modification uses a CNOT in place of a CZ when the central qubit is rotated in the Y or Z basis.Ancilla circuits target parity-calculating CNOT gates on one ancilla, allowing same-basis gates to move and cancel.Source: §6 · p. 31
• With basis-change-shift optimization, Bravyi-Kitaev and Jordan-Wigner were relatively equivalent for lexicographic and magnitude orderings; alternative orderings favored Bravyi-Kitaev but increased total gates for both.The Bravyi-Kitaev mapping did not substantially underperform Jordan-Wigner with this optimization.Source: §6 · pp. 31–32, Figure 10 · p. 32
• In ancilla circuits, Bravyi-Kitaev was slightly worse with magnitude ordering, but the difference disappeared at larger sizes, leaving no consistent mapping preference.For lexicographic ordering, the two mappings showed no predictable difference.Source: §6 · pp. 32–33
• Ancilla circuits showed up to around 30% gate reduction in the largest examples with lexoMag ordering, while their substantial savings could mask mapping-related gains or penalties.The authors caution that these circuits may be undesirable on architectures that lose nearest-neighbour CNOT chains.Source: §6 · pp. 32–33
• Random ordering produced the strongest Bravyi-Kitaev performance across the studied optimizations, but it was not a statistically meaningful sample of possible orderings.The authors describe the observed results as extremely consistent.Source: §6 · p. 33

7 Conclusions

• The study compares the mappings using advanced circuit-optimization techniques and reports substantially reduced gate-count estimates with Bravyi–Kitaev.The authors anticipate that applying the mapping could assist electronic-structure calculations on real quantum computers.Source: §7 · pp. 33, 35
• For systems involving more than 30 qubits, Bravyi–Kitaev dramatically reduces simulation expense, including for systems likely classically intractable to simulate.For approximately 50 qubits, the text reports a further gate-count reduction but does not provide its value in the supplied passage.Source: §7 · p. 33
• With optimized Trotter ordering, Jordan–Wigner can produce slightly shorter circuits for an individual Trotter step, while Bravyi–Kitaev is shorter in most cases.Figure 11 reports roughly equivalent performance for magnitude or lexicographic ordering; under an alternative ordering, Bravyi–Kitaev is superior, but by a greatly reduced extent.Source: §7 · pp. 33–34, Figure 11 · p. 34
• Bravyi–Kitaev is at worst roughly equivalent to Jordan–Wigner, and gate-count reductions are often particularly large for expensive entangling gates.The conclusion says it is superior in all cases aside from the lexicographic ordering.Source: §7 · pp. 34–35
• Bravyi–Kitaev typically reduces Trotter error for the studied orderings except magnitude ordering, with the difference approaching a factor of two in many larger lexicographic examples.The authors do not favor magnitude ordering despite its slightly reduced error, because lexicographic ordering has substantial overall gate-count savings.Source: §7 · p. 35
• The mapping’s performance depends on several factors; it is superior in most studied cases, but exceptions motivate numerical analysis across molecular systems, including classically intractable regimes.The authors also anticipate that applying the mapping could assist real-device electronic-structure calculations with computationally elusive results.Source: §7 · p. 35

ORCID

• Peter V. Coveney is listed with ORCID 0000-0002-8787-7256.Source: ORCID · p. 36

Funding

• The acknowledgments cite EPSRC studentship and grant support and an AFOSR award.The listed identifiers include EP/G037043/1, EP/L00030X/1, and FA9550-12-1-0046.Source: Funding · p. 36

Notes

• The authors declare no competing financial interest.Source: Notes · p. 36

A List of systems studied

• Table 2 covers all molecules studied and marks Trotterization-error analysis as performed (Y) or not performed (N).Source: Table 2 · p. 42

B Basis change shift

• The listed operations modify qubits in the original circuit, and each modified circuit remains equivalent in its end result.Source: B Basis change shift · p. 46