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Early fault-tolerant simulations of the Hubbard model

Earl T. Campbell

arXiv:2012.09238v4quant-ph

TL;DR

The paper addresses the cost and lattice-size limitations of early fault-tolerant Hubbard-model simulations based on split-operator Trotterization and the FFFT. It tightens Trotter error analysis and introduces plaquette Trotterization, obtaining lower phase-estimation Toffoli counts than prior split-operator estimates. The reported improvements are 5.5× at L = 8 and 7.8× at L = 16, with performance extending across even lattice sizes.

  • Problem

    Early fault-tolerant Hubbard simulations had strong resource estimates for SO-FFFT only on lattices with L = 2^k, while other lattice sizes incurred much higher costs.

    Source: §II · p. 3, §III · p. 4

  • Method

    The paper tightens split-operator commutator bounds using free-fermionic Hamiltonians and introduces plaquette Trotterization for square lattices.

    Source: §II · pp. 3–4, §III · p. 4

  • Results

    PLAQ improves on previous SO-FFFT T-counts by 5.5× at L = 8 and 7.8× at L = 16 while providing performance across all even L.

    Source: §V · p. 6

  • Takeaways & Limitations

    The results indicate a potentially useful fault-tolerant Hubbard-model application with resource requirements around one million Toffoli gates.

    Source: §V · p. 6, §VI · p. 7

Abstract

from arXiv · show

Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. A recent study of quantum algorithms for early simulations of the Hubbard model [Kivlichan \textit{et al.} Quantum 4 296 (2019)] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an $L \times L$ lattice with length $L=2^k$. On lattices with length $L \neq 2^k$, Givens rotations can be used instead of the FFFT but lead to considerably higher resource costs. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method, leading to $16 \times$ improvement in error bounds. Furthermore, we introduce plaquette Trotterization that works on any size lattice and apply our improved error bound analysis to show competitive resource costs. We consider a phase estimation task and show plaquette Trotterization reduces the number of non-Clifford gates by a factor $5.5\times$ to $9 \times$ (depending on the parameter regime) over the best previous estimates for $8 \times 8$ and $16 \times 16$ lattices and a much larger factor for other lattice sizes not of the form $L=2^k$. In conclusion, we find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.

I. THE HUBBARD MODEL HAMILTONIAN

The Hubbard Hamiltonian is split into a free-fermionic hopping term and an interacting term, with a shifted interaction form that preserves the spectrum up to a known electron-number-dependent constant.

Hamiltonian structure

  • The Hamiltonian is written as H = Hh + HI, where Hh contains hopping terms and HI contains interactions.The hopping component is free fermionic and exactly solvable through diagonalization of its coupling matrix.Source: §I · p. 2

Free-fermionic hopping

  • The hopping spectrum is symmetric about zero, and its operator norm satisfies ||Hh|| = ||R||1.Here ||R||1 is the Schatten 1-norm of the hopping matrix.Source: §I · p. 2

Shifted interaction

  • The shifted interaction Hamiltonian differs from the unshifted form by chemical-potential and identity shifts that commute with the Hamiltonian.Within a fixed η-electron subspace, the energy shift is the known constant −uη/2 + u/4 and can be removed classically.Source: §I · p. 2
  • Shifting the chemical potential tightens Trotter error bounds and reduces the gate count per Trotter step.The analysis assumes u > 0 throughout.Source: §I · p. 2

Target model

  • For the TDL, nearest-neighbour hopping has Ri,j = τ, while the difficult regime is 4 < u/τ < 12 and 10%–20% below half-filling.The TDL is the square-lattice specialization considered for much of the resource analysis.Source: §I · p. 2

II. IMPROVING THE SPLIT-OPERATOR ALGORITHM

The authors improve split-operator Trotterization by exploiting the shifted interaction and by evaluating commutator bounds through free-fermionic Hamiltonians, yielding tighter SO-FFFT resource estimates.

Trotter error

  • For a second-order Trotter step, the simulation error is bounded as ϵ ≤ Ws^3, where W depends strongly on the chosen decomposition.The split-operator decomposition uses HI and Hh as its two main terms.Source: §II · pp. 2–3

Prior split-operator method

  • A fast fermionic Fourier transform diagonalizes the hopping term efficiently on periodic square lattices with L = 2^k, whereas general Givens-rotation implementations require O(L^4) rotations.SO-FFFT previously achieved gate counts orders of magnitude below Givens rotations and other considered approaches.Source: §II · p. 3

SO-FFFT+ improvements

  • SO-FFFT+ reduces interaction-implementation cost from 3L^2 to L^2 arbitrary Z-axis rotations by using the shifted Hamiltonian.The shifted interaction maps to a single Z⊗Z Pauli operator per site under Jordan–Wigner.Source: §II · p. 3
  • The new commutator analysis proves WSO ≤ min[WSO1, WSO2] for the TDL and improves the resulting error-bound estimates by an average factor of 16×.The two bounds correspond to the two possible orderings of the split-operator decomposition.Source: §II · p. 4

Commutator analysis

  • The improved bounds are obtained by bounding nested commutators involving HI and Hh through free-fermionic Hamiltonians whose norms and commutators can be efficiently computed.This avoids the substantial loosening caused by repeated triangle-inequality expansions into Pauli terms.Source: §II · pp. 3–4

III. PLAQUETTE TROTTERIZATION

Plaquette Trotterization partitions square-lattice hopping edges into disjoint plaquette layers, providing a second-order decomposition that extends beyond power-of-two lattice sizes.

Plaquette construction

  • PLAQ partitions square-lattice edges into pink and gold sets so that each edge belongs to exactly one plaquette, then uses HI, Hp h, and Hg h as Trotter terms.The construction is possible for even L and can be modified for odd L using plaquettes with fewer edges.Source: §III · p. 4

Lattice-size scope

  • PLAQ avoids the FFFT restriction to L = 2^k and is designed to operate on other lattice sizes.This broadens the lattice sizes accessible without resorting to the much costlier general Givens-rotation route.Source: §III · p. 4

Error analysis

  • The additional PLAQ commutator terms are free fermionic and efficiently computable, producing only a slight increase in error bounds relative to SO-FFFT+.The plaquette error analysis reuses the split-operator bound together with additional free-fermionic terms.Source: §III · pp. 4–5

Implementation

  • Each plaquette Hamiltonian acts on four fermions and can be implemented with 8 T gates and 2 equal-magnitude Z-axis rotations.There are L^2/4 plaquettes per colour and two spin sectors.Source: §III · p. 5
  • For a half plaquette evolution, the implementation requires 4L^2 T gates and L^2 Z-axis rotations, while the full Trotter step uses two such plaquette evolutions.The interaction evolution is added separately in the full step.Source: §III · p. 5

IV. ROTATION SYNTHESIS

The rotation-synthesis analysis compares standard arbitrary-angle synthesis with Hamming weight phasing, which particularly benefits PLAQ because its rotation angles are uniform.

Standard synthesis

  • Standard synthesis can require 10–50 T gates per arbitrary-angle Z rotation, making raw rotation counts important for total fault-tolerant cost.Under this accounting, SO-FFFT+ can have a lower T-count on applicable power-of-two lattices despite PLAQ’s lower per-step rotation count.Source: §IV · p. 5

Hamming weight phasing

  • Hamming weight phasing replaces many equal-angle rotations with ⌈log2(m) + 1⌉ rotations on an ancillary Hamming-weight register.The method uses α additional ancillas and α Toffoli gates, equivalent to 4α T gates.Source: §IV · p. 5
  • For large m, Hamming weight phasing reduces the effective cost of an arbitrary rotation to approximately 4 T gates, with α = m − 1 for m = 2^k.For general m, α ≤ m − 1, and tighter bounds use α = m − w(m).Source: §IV · p. 5

Method comparison

  • PLAQ is especially compatible with Hamming weight phasing because its interaction rotations all have equal angle, whereas SO-FFFT+ has eigenvalue-dependent angles.Consequently, PLAQ can exploit Hamming weight phasing more comprehensively than SO-FFFT+.Source: §IV · pp. 5–6

Resource accounting

  • Table II resource estimates use α = L^2/2 Hamming-weight ancillas and include phase-estimation and repeat-until-success qubits in the logical-qubit count.A complete architecture-specific estimate additionally requires error-correction, factory, routing, and catalyst-storage overheads.Source: Table II · p. 6

V. PHASE ESTIMATION

The phase-estimation analysis relates Trotter error to effective-Hamiltonian error and optimizes the simulation time against precision costs. Plaquette Trotterization becomes increasingly advantageous with available ancillas and improves prior split-operator estimates across even lattice sizes.

V. PHASE ESTIMATION

  • Optimizing the phase-estimation simulation time balances Trotter error against the increased precision and gate usage caused by smaller time steps.The analysis assumes Wt^3 ≪ 1 and bounds the effective-Hamiltonian error by Wt^2.Source: §V · p. 6
  • The phase-estimation success probability equals the approximate ground-state overlap f, while state preparation itself is left unaddressed.An ideal implementation returns the ground-state energy with probability f, up to additive error Wt^2.Source: §V · p. 6
  • The resource comparison reports Toffoli counts by converting T-gates through N_TOF + (N_T/2), motivated by low-overhead Toffoli-state factories.The paper uses Toffoli gates rather than T-gates for the resource comparison.Source: §V · p. 6
  • Increasing available HWP ancillas makes PLAQ outperform SO-FFFT+ in T-count.With few ancillas, SO-FFFT+ instead has lower T-count.Source: §V · p. 6
  • A factor 5.5× improvement at L = 8 and 7.8× at L = 16 over previous SO-FFFT T-counts extends across all even L.Givens rotations and fermionic swap networks work for arbitrary L but have orders-of-magnitude higher Toffoli counts.Source: §V · p. 6
  • With a constant fraction of qubits allocated to HWP, PLAQ and SO-FFFT+ show comparable performance as L varies.The comparison is made by varying L while maintaining that allocation rule.Source: §V · p. 6

VI. CONCLUSIONS

The conclusions place Hubbard-model simulation among potential early fault-tolerant applications and attribute improved prospects to tighter error bounds and lower gate complexity. The appendices support this analysis with free-fermion spectral facts and commutation properties used in the derivations.

VI. CONCLUSIONS

  • The paper concludes that tighter error bounds and reduced gate complexity improve the prospects for Hubbard simulation on an early fault-tolerant quantum computer.Source: §VI · p. 7

Appendix A: Free fermionic Hamiltonians

  • Free-fermionic Hamiltonians can be diagonalized through their mode-space matrix representation, whose diagonal entries give the canonical-mode energies.The spectrum is obtained from occupied subsets of these modes, with the ground state filling the lowest-energy states.Source: Appendix A · p. 8
  • For hopping Hamiltonians, the operator norm is characterized using the associated coefficient matrix and its Schatten 1-norm.Source: Appendix A · p. 9

Appendix B: Anti-commutation

  • The appendices establish that the relevant fermionic operators obey commutation or anti-commutation relations that support the nested-commutator analysis.In particular, the interaction operator anticommutes with the hopping terms involving the same site pair.Source: Appendix A · p. 9, Appendix B · p. 9
  • The anti-commutator calculation yields zero for the displayed operator relation, confirming the algebra used later in the error-bound derivation.Source: Appendix B · p. 9

1. Overview

The overview develops tighter second-order Trotter error bounds for the Hubbard model and reports substantially improved bounds over prior numerical estimates. The analysis uses commutator norms and compares the two possible second-order orderings.

1. Overview

  • Theorem 1 provides second-order Trotter error bounds for the Hubbard model on an L × L periodic square lattice.Source: §1 · pp. 9–10
  • The new bounds are about 16 times better than Kivlichan et al. for u/τ = 4 and 8.Because the shifted Hamiltonian changes the Trotter sequence, the comparison is technically between different sequences; W_SO1 is best unless u ≫ τ.Source: §1 · p. 10
  • The bounds are obtained from nested commutator norms involving H_h and H_I, with the smaller of two expressions chosen because their gate counts are essentially equal.Source: §1 · p. 10
  • Lemmas 1 and 2 plus Corollary 1 bound the nested commutators needed to obtain W_SO1 and W_SO2.Source: §1 · p. 10

2. First lemma for Trotter error bounds

The first-lemma analysis exploits the algebra of Hubbard interaction and hopping operators to evaluate nested commutators. It then uses free-fermionic structure and precomputed lattice norms to make the resulting bounds computable.

2. First lemma for Trotter error bounds

  • The lemma proof begins from the Hubbard interaction and hopping decomposition and uses notation for the operators and coefficient matrices throughout.Source: §2 · p. 10
  • The commutator [H_I, H_h] is simplified by the fact that interaction terms commute with unrelated bonds and anti-commute with terms sharing a site.Source: §2 · p. 10
  • Unitary conjugation and operator-norm properties transform the nested-commutator expression into a tractable bound.Source: §2 · p. 10
  • Because H_h is free fermionic, its energy and norm can be obtained from the associated coefficient matrix, completing the first-lemma bound.Source: §2 · p. 10
  • For periodic L × L square lattices, Table III supplies normalized hopping norms and nested-commutator values for direct evaluation of the bound.The table notes ∥H_h∥/τ ≤ (3/2)L^2 for L ≥ 4.Source: Table III · p. 11

3. Second lemma for Trotter error bounds

The second lemma bounds nested commutators involving local interaction terms and the hopping Hamiltonian by exploiting their free-fermionic structure. For the square-lattice Hubbard model, these bounds yield lattice-size-independent intermediate quantities and a specialized corollary.

Second lemma for Trotter error bounds

  • The lemma evaluates nested commutators by expanding [PQ, R] and applying operator-norm bounds, triangle inequalities, and anticommutation relations.The derivation rewrites the interaction term and the local hopping operator before symmetrizing the resulting expressions.Source: §3 · p. 11
  • The general lemma is specialized to the TDL Hubbard model on an L × L periodic square lattice to obtain a corollary for the relevant nested commutator.The proof concludes after substituting the norm estimates into the nested commutator expression.Source: §3 · pp. 11–12
  • For the square lattice, the local hopping operator satisfies ||T_i|| = ||R_star||_1 = 4τ.This value enters the specialized evaluation of the nested-commutator bound.Source: §3 · p. 12
  • The commutator [T_i, H_h] is represented by the free-fermionic coefficient matrix [R_star, R], making its norm efficiently computable.Because R_star acts locally on five electrons, the resulting norm is independent of lattice size once L is sufficiently large.Source: §3 · p. 12

Appendix D: PLAQ Trotter error

PLAQ partitions hopping terms into regular plaquette-based components, extending second-order Trotterization beyond the lattice sizes supported by FFFT. Its local structure permits explicit circuits and resource estimates that are competitive with FFFT and lower than Givens rotations.

Appendix D: PLAQ Trotter error

  • PLAQ applies to any even lattice size and has gate counts comparable or superior to FFFT while remaining far below Givens-rotation costs.The construction partitions square-lattice edges into pink and gold sets, with odd-size lattices omitted from the detailed treatment.Source: Appendix D · p. 12
  • The PLAQ error bound combines standard second-order nested-commutator terms, with the first two matching the previously calculated split-operator contributions.The remaining symmetric commutators represent the excess error introduced by plaquette layering.Source: Appendix D · pp. 12–13
  • The nested commutators for plaquette hopping components are free-fermionic and can be evaluated efficiently from their coefficient matrices.For periodic square lattices, the tabulated norm values are upper bounded by 10 across the stated range.Source: Appendix D · p. 13

Appendix E: Realising plaquette Trotterization

  • Each plaquette unitary uses two arbitrary Z rotations and four F gates, with each F gate requiring two T gates.The resulting per-plaquette and tile costs are given as N_plaq = {0, 8, 2} and N_tile = {0, 4L^2, L^2}.Source: §1 · pp. 13–14

1. Resource cost without ancilla

  • Without ancilla, the total plaquette resource vector is {0, 12L^2, 4L^2}, while Hamming-weight phasing can reduce the cost of repeated rotations.The resource model counts Toffoli gates, T gates, and arbitrary single-qubit Z rotations per Trotter step.Source: §1 · pp. 13–14

2. Using ancilla and Hamming weight phasing

Hamming weight phasing reduces repeated Z-rotation costs by computing the binary Hamming weight into ancillas and applying rotations conditioned on that value. The paper formalizes the resulting Toffoli and ancilla requirements and applies the method to PLAQ layers.

2. Using ancilla and Hamming weight phasing

  • Hamming weight phasing implements m equal-angle phase gates using ⌊log2(m) + 1⌋ arbitrary rotations, α Toffoli gates, and α clean ancillas.The construction computes and uncomputes the Hamming weight of the relevant bit string.Source: §2 · p. 14
  • The method refines an earlier Hamming-weight-phasing construction by using the binary weight w(m) to characterize its multiplicative complexity.This connects the circuit cost to the number of ones in m’s binary expansion.Source: §2 · p. 14
  • For PLAQ, repeated layers containing L^2 or L^2/2 identical rotations can be batched whenever m divides L^2/2.The resulting cost transformation assumes α = m − w(m) logical ancillas.Source: §2 · p. 14
  • The ancilla trade-off is explicit: Hamming weight phasing lowers T-count but requires additional logical ancillas and Toffoli gates.The worst case has α = m − 1, while non-power-of-two values can make the bound looser than this worst case.Source: §2 · p. 14

Appendix F: Phase estimation overview

The phase-estimation analysis allocates a target energy error among Trotterization, phase-estimation, and rotation-synthesis errors, then optimizes the Trotter step and circuit costs. For the Hubbard model, the remaining free error-split parameter is numerically near x ∼ 0.01.

Appendix F: Phase estimation overview

  • Phase estimation has three additive energy-error sources: Trotter error, phase-estimation error, and synthesis error for single-qubit Z rotations.The total target error requires ΔH_T + δ ≤ ϵ, with δ covering the Trotter and phase-estimation components.Source: Appendix F · p. 15
  • The Trotter and phase-estimation errors are optimized through the Trotter step t, while synthesis error is controlled through the T-count assigned to each arbitrary rotation.The analysis differentiates the combined Trotter and phase-estimation contribution with respect to t.Source: Appendix F · p. 15
  • For a target error ϵ, the analysis splits the synthesis and remaining error budgets as ΔH_T = xϵ and δ = (1 − x)ϵ.This leaves x as the free parameter after fixing a circuit implementation.Source: Appendix F · p. 15
  • Numerically, x ∼ 0.01 is a good choice for the Hubbard model, and the resulting total T-gate cost can be converted to a Toffoli count using catalysis.The optimum is reported for the specified Hubbard-model calculations rather than as a universal constant.Source: Appendix F · p. 15
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