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Simulating chemistry efficiently on fault-tolerant quantum computers
N. Cody Jones, James D. Whitfield, Peter L. McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, Yoshihisa Yamamoto
TL;DR
Quantum chemistry calls for accurate simulation, but quantum algorithms need efficient implementations to be useful. This paper develops fault-tolerant gate constructions and analyzes chemistry simulation algorithms, estimating a LiH ground-state calculation at ∼5.6 hours under a specified hypothetical gate-time assumption.
Problem
Quantum chemistry needs accurate simulation beyond approximate classical methods, while quantum simulation must be implemented efficiently to be useful as a scientific tool.
Source: §1 · p. 2Method
The paper constructs efficient fault-tolerant arbitrary phase rotations and analyzes first- and second-quantized chemistry simulation algorithms.
Source: §1 · pp. 2–3, §2 · p. 4Results
∼5.6 hours is the estimated LiH ground-state simulation time on a hypothetical fault-tolerant computer with 1 ms per error-corrected gate.
Source: §1 · p. 2Takeaways & Limitations
LiH retains features of more complex chemistry simulations while offering a compact problem for experimental demonstrations, though it does not itself demonstrate quantum advantage.
Source: §1 · p. 3Constant-depth first-quantized Hamiltonian evaluation requires a significantly larger quantum computer.
Source: §4.3 · p. 24
Abstract
from arXiv · showhide
Quantum computers can in principle simulate quantum physics exponentially faster than their classical counterparts, but some technical hurdles remain. Here we consider methods to make proposed chemical simulation algorithms computationally fast on fault-tolerant quantum computers in the circuit model. Fault tolerance constrains the choice of available gates, so that arbitrary gates required for a simulation algorithm must be constructed from sequences of fundamental operations. We examine techniques for constructing arbitrary gates which perform substantially faster than circuits based on the conventional Solovay-Kitaev algorithm [C.M. Dawson and M.A. Nielsen, \emph{Quantum Inf. Comput.}, \textbf{6}:81, 2006]. For a given approximation error $\epsilon$, arbitrary single-qubit gates can be produced fault-tolerantly and using a limited set of gates in time which is $O(\log \epsilon)$ or $O(\log \log \epsilon)$; with sufficient parallel preparation of ancillas, constant average depth is possible using a method we call programmable ancilla rotations. Moreover, we construct and analyze efficient implementations of first- and second-quantized simulation algorithms using the fault-tolerant arbitrary gates and other techniques, such as implementing various subroutines in constant time. A specific example we analyze is the ground-state energy calculation for Lithium hydride.
1. Introduction
The introduction motivates fault-tolerant quantum chemistry as a route to exact molecular simulation and focuses on efficient simulated-time evolution. It presents constructive gate and algorithm methods, with LiH as a concrete resource-analysis example.
1. Introduction
- Quantum chemistry is motivated by applications including molecular-Hamiltonian eigenvalues, while commonly employed classical methods remain approximate.The introduction notes that chemistry and band-structure calculations account for up to 30% of computation time at supercomputer centers.Source: §1 · p. 2
- Quantum simulators are predicted to enable exact chemical simulation within a basis, and a 128-logical-qubit computer was projected to outperform classical computers.The introduction connects this prospect to high-quality ab initio data and analysis of complex chemical mechanisms.Source: §1 · p. 2
- ∼5.6 hours is the estimated LiH ground-state simulation time on a hypothetical fault-tolerant computer with 1 ms per error-corrected gate.The paper uses this example to argue that practical execution times can be analyzed quantitatively.Source: §1 · p. 2
- The paper focuses on fault-tolerant circuit-model simulation and efficient constructions of arbitrary gates, whose execution time affects simulation performance.Its broader plan covers phase rotations, second- and first-quantized algorithms, resource analysis, and selecting parameters under engineering constraints.Source: §1 · pp. 2, 4
- Digital simulation has three steps—state preparation, simulated-time evolution, and readout—and this paper focuses on the evolution step.The system is evolved in time steps; state preparation and measurement readout are necessary but not discussed here.Source: §1 · p. 3, Figure 1 · p. 3
- LiH ground-state energy calculation is used to illustrate the proposed methods while retaining features of more complex chemistry simulations.The compact problem is also presented as convenient for experimental demonstrations, although it does not itself demonstrate quantum advantage.Source: §1 · p. 3
2. Fault-tolerant phase rotations
The section develops fault-tolerant ways to implement arbitrary phase rotations from a finite gate set and compares their resource costs. Programmable ancilla rotations can reduce average circuit depth through advance, parallel ancilla preparation, while requiring excess qubits.
2.1. Phase kickback
- Phase kickback adds an integer to an ancilla-register eigenstate, imprinting a phase on the control qubit; accuracy and resource costs depend on the ancilla-state size.The controlled addition is approximately equivalent to an arbitrary RZ(φ) rotation and uses fault-tolerant gate primitives and ancillas.Source: §2.1 · pp. 4–5, Figure 2 · p. 5
2.2. Gate approximation sequences
- Gate-approximation sequences compile arbitrary rotations into fault-tolerant single-qubit gates; Solovay-Kitaev has poly(log ε) asymptotic length, while Fowler offers shorter sequences but greater classical complexity.Both methods trade more quantum gates for better approximation accuracy.Source: §2.2 · p. 6
2.3. Programmable ancilla rotation
- Precomputed ancillas let PAR achieve constant average depth for arbitrary rotation accuracy with sufficient parallel preparation, but the approach favors computers with many excess qubits.Each round succeeds with 50% probability; with M prepared ancillas, the probability of needing a deterministic fallback is 2^-M.Source: §2.3 · pp. 6–7, Figure 4 · p. 7
2.4. Analysis of a single-qubit phase rotation
- Solovay-Kitaev sequences cost substantially more in circuit depth and T gates; for ε ≤10^-6, phase kickback may be preferred to Fowler sequences.Fowler sequences are compact, but their classical calculation takes about a day per rotation at ε ≤10^-4; phase kickback has comparable quantum resources and uses adders that are trivial to compile. The analysis assumes ideal logical gates and full parallelism, excluding physical error-correction overhead.Source: §2.4 · pp. 8, 10, Figure 5 · p. 9
3. Simulating chemistry in second-quantized representation
The second-quantized approach expresses the electronic Hamiltonian using creation operators and precomputed integrals, then implements its evolution with Jordan-Wigner transforms and controlled phase rotations. The section improves these circuit components and analyzes their resource costs for LiH.
3.1. Controlled phase rotations
- Controlled rotations use an ancilla and one phase rotation; Fowler sequences are about three times faster than phase kickback, while controlled-PARs average four gates.PARs shift computation to precomputed ancillas, so their preparation still requires another rotation method.Source: §3.1 · p. 12, Figure 9 · p. 13
3.2. Finite precision in pre-calculated integrals
- At a 10^-10 atomic-unit cutoff, STO-3G retains 99 of 231 nonzero integrals and TZVP retains 10,315 of 22,155; a 10^-4 TZVP cutoff leaves just over 9,000 terms.Raising the cutoff reduces circuit work but affects simulation accuracy.Source: §3.2 · p. 14, Figure 10 · p. 14
3.3. Jordan-Wigner transform using teleportation
- Teleportation rearranges the Jordan-Wigner CNOT ladder for constant-time implementation, with Bell-state ancillas and Pauli frames tracking teleportation errors.The approach can extend to other Clifford circuits if enough Bell states are available.Source: §3.3 · p. 15, Figure 11 · p. 15
3.4. Resource analysis for ground-state energy simulation of LiH
- For 1,023 simulated time steps, the LiH simulation takes ∼5.6 hours using PARs versus ∼3.8 years using Solovay-Kitaev rotations.The estimate assumes a 1 ms duration per quantum gate and corresponds to a 10-bit energy estimate.Source: §3.4 · p. 16, Figure 12 · p. 17
- PARs run faster but consume more T gates than Fowler sequences or phase kickback, making the choice dependent on quantum-computer capabilities.At rotation accuracy ϵ ≤10^-4, PARs do not account for most circuit depth, whereas Solovay-Kitaev rotation depth is too large to show to scale.Source: §3.4 · p. 16, Figure 13 · p. 18
4. Simulating chemical structure and dynamics in first-quantized representation
The section develops first-quantized molecular simulation using grid-based wavefunctions, quantum variable rotations, and parallel evaluation of potential energy. Its resource analysis highlights trade-offs between circuit depth and qubit count, including for LiH.
4. Simulating chemical structure and dynamics in first-quantized representation
- First-quantized simulation encodes particles on a Cartesian grid and alternates kinetic- and potential-energy evolution using a split-operator approximation.The kinetic operator is diagonal in momentum basis, reached from position basis with quantum Fourier transforms along each spatial dimension.Source: §4 · pp. 18–20
4.1. Quantum variable rotation
- The bitwise QVR applies a separate rotation to each register qubit, requiring each rotation’s accuracy to be ε/t for a t-bit QVR with total accuracy ε.The phase-kickback alternative uses an addition circuit with a programmed ancilla register; registers can be reused when scaling factors recur, while PAR-ancilla preparation pays off only for frequently needed angles.Source: §4.1 · pp. 20–23
- Phase-kickback QVRs use substantially fewer T gates than bitwise rotations at comparable circuit depth.The single phase rotation also need not be as accurate as each bitwise rotation to achieve the same total QVR accuracy.Source: §4.1 · p. 22, Figure 16 · p. 22
- Phase-kickback QVRs can replace blocks of controlled rotations in fault-tolerant quantum Fourier transforms, reducing T-gate use compared with individually implemented rotations.The method also applies to approximate QFTs by truncating the register size.Source: §4.1 · p. 23
4.2. Improved parallelism in potential energy operator
- Because Coulomb terms commute, disjoint particle pairs can be evaluated together, reducing O(b^2) pair interactions to O(b) time.The potential operator is diagonal in the position basis, so its terms can be calculated in any order.Source: §4.2 · p. 23
- Copying position-register data enables all particle pairs to be evaluated in constant time, at the cost of O(b^2) memory and circuit effort.The copies remain entangled with one another; the approach is not cloning.Source: §4.2 · pp. 23–24
4.3. Resource analysis for first-quantized molecular simulations
- First-quantized simulation’s errors are systematically improvable by increasing spatial and temporal precision, while constant-depth Hamiltonian evaluation requires a substantially larger quantum computer.The constant-depth implementation is described as probably best suited to large-scale quantum computers.Source: §4.3 · p. 24
- At six particles, first-quantized depth is comparable to PAR-based second-quantized simulation but requires many more qubits; its depth is asymptotically lower as particle count grows.Figure 17 assumes 10 qubits per spatial dimension and 1023 time steps, corresponding to about 3 significant figures.Source: §4.3 · p. 24, Figure 17 · p. 25
5. Comparing simulation methods
The section compares implementations through efficient frontiers of qubit count and circuit depth, with the preferred method depending on available resources and the chosen cost function. For LiH, compact Fowler and faster PAR implementations trade qubits for speed, while first-quantized simulation may benefit more at larger particle counts or for time-varying dynamics.
5. Comparing simulation methods
- An efficient frontier contains the minimum achievable depth for each machine size, and the best implementation depends on the chosen cost function.The cost function can combine estimated engineering challenge with execution-time penalties.Source: §5 · pp. 25–26
- For LiH, Fowler sequences offer a compact option and PAR a faster one that requires additional ancilla qubits; first-quantized simulation may gain more for larger particle counts or time-varying dynamics.Figure 18 plots efficient frontiers for implementations using qubit count and circuit depth; the PAR and first-quantized frontiers allow depth reduction by increasing application qubits.Source: §5 · p. 26, Figure 18 · p. 27
- The comparison does not include comprehensive system costs such as error correction and non-Clifford-gate production, and future algorithms could change the frontiers.Source: §5 · p. 26
6. Conclusions
The conclusions compare fault-tolerant phase-rotation methods and identify distinct resource improvements for second- and first-quantized simulation. The methods may also extend to other Hamiltonian simulations and quantum algorithms requiring Fourier transforms.
6. Conclusions
- Phase kickback offers rotation circuits of depth O(log ϵ) or O(log log ϵ), while programmable ancilla rotations reduce depth further by precomputing ancillas.Fowler sequences are shortest, but classical sequence generation becomes intractable for high precision such as ϵ < 10^-6.Source: §6 · p. 27
- Second-quantized simulation uses fewer qubits but has asymptotically longer circuit depth than first-quantized simulation as particle count grows.The paper proposes integral thresholding, constant-time Jordan-Wigner transforms, and PARs to reduce second-quantized resource costs.Source: §6 · pp. 27–28
- First-quantized improvements include QVRs with arbitrary scaling and parallel potential-energy calculations taking linear time without extra qubits or constant time with quadratically scaling qubits.The constant-time option grows quadratically rather than linearly in qubit count with the number of simulated particles.Source: §6 · p. 28
- Fault-tolerant rotations may also apply to spin lattices, lattice gas automata, lattice gauge theories, quantum chaos, and algorithms requiring Fourier transforms.The authors present the methods as a flexible set for making simulation algorithms practically realizable on fault-tolerant quantum computers.Source: §6 · p. 28
Appendix A. Transforming the phase kickback register
The appendix constructs a phase-kickback ancilla transformation that avoids a costly modular-multiplication step. It builds the transformation from separable ancilla states and corrective phase rotations.
Appendix A. Transforming the phase kickback register
- The proposed ancilla state avoids solving Eqn. (4) in a quantum circuit, which would otherwise require an expensive multiplication operation.The authors propose the construction as a way to avoid operations associated with modular multiplication.Source: Appendix A · p. 28
- The ancilla register states are separable, and the transformation uses single-qubit phase rotations with a controlled-addition circuit.The first bit is always 1 when k is odd.Source: Appendix A · p. 29
- Corrective rotations on successive bits complete the transformation in circuit depth O(n^2) gates or less, depending on the phase-kickback adder.The correction for the mth bit uses the preceding m−1 ancilla bits and phase kickback.Source: Appendix A · p. 29
Appendix B. Quantum circuits for potential and kinetic energy operators in first-quantized molecular Hamiltonians
The appendix describes digital arithmetic for first-quantized molecular energy operators on a Cartesian-grid wavefunction. It details potential-energy evaluation through reciprocal-distance calculations and QVRs, alongside the kinetic-energy calculation.
Appendix B. Quantum circuits for potential and kinetic energy operators in first-quantized molecular Hamiltonians
- First-quantized simulation represents the wavefunction on a Cartesian grid, where digital arithmetic acts on coordinate registers and the potential is a sum of Coulomb interactions.The potential operator is diagonal in the position basis.Source: Appendix B · p. 29
- For each particle pair, the potential calculation requires 1/|r_i−r_j|; multiplication uses quantum adders, and the reciprocal is calculated with Newton-Raphson iteration.The pairwise Coulomb prefactor can be encoded into the QVR scaling factor.Source: Appendix B · p. 29
- The Newton-Raphson calculation converges within 5 iterations at 32-bit arithmetic with a suitably chosen initial value.Typically, less precision is required for simulation.Source: Appendix B · p. 29
- The QVR kicks back a phase to the position-basis wavefunction, and uncomputing intermediate registers implements the potential-energy propagator.The QVR scaling factor includes the particle-pair charges and the evolution time step.Source: Appendix B · p. 30
- A quantum Fourier transform maps each spatial dimension from position to momentum basis, and the kinetic energy is the sum of individual particle kinetic-energy operators.The momentum representation permits direct calculation of the wavevector magnitude squared.Source: Appendix B · p. 30