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Efficient discrete-time simulations of continuous-time quantum query algorithms
R. Cleve, D. Gottesman, M. Mosca, R. D. Somma, D. L. Yonge-Mallo
TL;DR
The paper addresses the lack of a general efficient conversion from continuous-time to discrete quantum query algorithms, especially for arbitrary driving Hamiltonians. It develops a simulation based on fractional queries, low-Hamming-weight control states, and error correction, achieving O(T log T / log log T) discrete queries for continuous query time T. Consequently, discrete-query lower bounds transfer to continuous time up to a sub-logarithmic factor, although the error dependence for arbitrary initial quantum states remains a limitation.
Problem
No general efficient conversion was known from continuous-time query algorithms to discrete queries, despite the continuous-time model's usefulness for algorithm discovery.
Source: §I · p. 2Method
The paper simulates continuous evolution using fractional queries, probabilistic controlled full queries, low-Hamming-weight control states, and recursive correction of post-selection errors.
Source: §I · p. 2, A. Rough overview of the proof of… · p. 3Results
O(T log T / log log T) discrete queries simulate any continuous-time algorithm of total query time T, independently of the driving Hamiltonian norm.
Source: §I · p. 2Takeaways & Limitations
A discrete-query lower bound of T yields a continuous-time lower bound of Ω(T log log T / log T), making the models equivalent up to a sub-logarithmic factor.
Source: §I · p. 2For arbitrary initial quantum states, the approach requires a factor of ε^-1 for failure probability ε, although the authors conjecture this may improve to O(log(1/ε)).
Source: §III · p. 11
Abstract
from arXiv · showhide
The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in the continuous-time query model whose total query time is T can be simulated by a quantum algorithm in the discrete query model that makes O[T log(T) / log(log(T))] queries. This is the first upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of \Omega[T log(log(T))/log (T)] in the continuous-time query model.
I. INTRODUCTION AND SUMMARY OF RESULTS
The paper frames continuous-time queries as a useful but not generally efficiently convertible model, then establishes a driving-operation-independent simulation by discrete queries. This yields near-equivalence between the models up to a sub-logarithmic factor.
I. INTRODUCTION AND SUMMARY OF RESULTS
- The continuous-time model evolves an initial state under the input query Hamiltonian plus an arbitrary time-dependent driving Hamiltonian before measurement.The algorithm is specified by D(t), an initial state, an execution time T, and a final measurement.Source: §I · pp. 1–2
- Continuous-time queries have supported algorithm discovery, but previous conversions lacked a general efficient bound independent of driving operations.Earlier approaches could incur costs depending strongly on the driving Hamiltonian norm.Source: §I · p. 2
- The result answers Mochon's question by showing that continuous-time queries cannot generally circumvent discrete-query lower bounds.Known examples such as search and parity already showed no asymptotic continuous-time advantage, but the general case was open.Source: §I · p. 2
- Continuous-time query algorithms can be simulated with O(T log T / log log T) discrete queries, regardless of the driving Hamiltonian's norm.The bound applies even when the driving Hamiltonian is highly energetic.Source: §I · p. 2
- A discrete lower bound of T queries implies a continuous-time lower bound of Ω(T log log T / log T), while T discrete queries can be simulated continuously in O(T) time.Thus the models are equivalent up to a sub-logarithmic factor.Source: §I · p. 2
A. Rough overview of the proof of Theorem 1
The proof overview converts continuous evolution into many fractional queries, implements those queries probabilistically, compresses their controls to low Hamming weight, and recursively corrects post-selection failures. This organization reduces the expected discrete-query cost to O(T log(T/ε)/log log(T/ε)).
A. Rough overview of the proof of Theorem 1
- A direct Trotter-Suzuki discretization can require O(∥D∥T^2/ε) full queries, so it is inefficient when the driving Hamiltonian norm is large.Using full queries for every small-time evolution also creates an unfavorable dependence on ∥D∥.Source: A. Rough overview of the proof of… · pp. 2–3
- Each small-time query evolution is replaced by a controlled discrete query with a low-amplitude control state, succeeding conditionally with probability approximately 1 − T/p.Because roughly T of p fractional queries may fail, the proof needs explicit error correction.Source: A. Rough overview of the proof of… · p. 3
- The control qubits are approximated by states supported on low Hamming weight, letting the circuit be truncated to O(m(T/p) log(T/ε)/log log(T/ε)) full queries per segment.The Hamming weight corresponds to the number of full queries, so truncation removes atypical high-query branches.Source: A. Rough overview of the proof of… · p. 3
- Post-selection measurements reveal error locations, enabling failed segments to be undone and retried recursively while increasing the expected number of segments only by a constant factor.The resulting simulation uses O(T) segment simulations and expected query cost O(T log(T/ε)/log log(T/ε)).Source: A. Rough overview of the proof of… · p. 3
- The proof is organized into fractional-query conversion, probabilistic controlled-query simulation, low-Hamming-weight approximation, error correction, and fidelity analysis.These stages culminate in the theorem's discrete-query complexity bound.Source: A. Rough overview of the proof of… · pp. 2–3
II. DISCRETE QUERY SIMULATION OF CONTINUOUS-TIME QUERY ALGORITHMS
The simulation discretizes continuous query evolution into fractional queries whose total fractional cost tracks the original query time, then selectively realizes them with full discrete queries. Omitting many full queries preserves high fidelity while avoiding dependence on the number of fractional steps.
II. DISCRETE QUERY SIMULATION OF CONTINUOUS-TIME QUERY ALGORITHMS
- A fractional query applies a phase generated by H_x for angle θ, with cost |θ|/π; θ = π gives a full query.A fractional-query algorithm alternates these operations with driving unitaries and sums their fractional costs.Source: §II · p. 3
- A continuous-time algorithm of query time T can be approximated by fractional queries of total cost T/π, even though their count p may be much larger than T.Naively replacing every fractional query by two full queries would therefore create excessive overhead.Source: §II · p. 3
- The construction reorganizes driving and query operations so many full queries can be omitted while retaining constant success probability and very high output fidelity.This is the central mechanism for reducing the discrete-query overhead.Source: §II · p. 3
- The resulting full-query cost is O(T log T / log log T), independent of the number of fractional queries and the approximation parameter in the naive construction.The construction is designed to avoid the potentially much larger overhead from directly simulating every fractional query.Source: §II · p. 3
- A fractional query can be implemented using two controlled full queries, with an ancilla operation applying the desired phase to the |−⟩ state.The figure illustrates this two-query controlled construction.Source: Figure 1 · p. 4, §II · p. 4
A. Converting a continuous-time query algorithm to a fractional query algorithm
The paper approximates continuous-time evolution with fractional queries using a time-dependent Trotter construction, while controlling the dependence on the driving Hamiltonian’s average norm. Although the fractional-query count can grow with that norm, the eventual discrete-query cost is independent of it.
A. Converting a continuous-time query algorithm to a fractional query algorithm
- The time-dependent Trotter-Suzuki construction approximates a continuous-time algorithm of query time T with fractional-query cost T/π.The construction applies to arbitrary precision ε > 0 and uses the driving Hamiltonian’s average norm in choosing the discretization.Source: A. Converting a continuous-time query… · p. 4
- For fidelity 1−ε1, decomposing [0,T] into p ≥ 2T^2r/√ε1 intervals gives fractional queries of size θ = T/p.Here r is an upper bound on the integrated norm of D(t), and the fractional-query algorithm alternates driving and query evolution.Source: A. Converting a continuous-time query… · pp. 4–5
- The resulting fractional-query circuit has total fractional cost T/π but may contain a much larger number of fractional queries as r increases.The circuit approximation is shown in Fig. 2, with each interval contributing a fractional query.Source: A. Converting a continuous-time query… · p. 5, Figure 2 · p. 5
- Replacing each fractional query directly by two full queries can create an undesirable overhead when the number of fractional queries greatly exceeds T.The full construction avoids this overhead and achieves O(T log T/ log log T) discrete queries independent of r and ε1.Source: A. Converting a continuous-time query… · p. 5
B. Simulating fractional queries with low amplitude controlled discrete queries
Each fractional query is probabilistically implemented with a low-amplitude controlled full query and an ancilla measurement. The successful branch applies the desired fractional query, while the failure branch produces a correctable error operation.
B. Simulating fractional queries with low amplitude controlled discrete queries
- A controlled full query followed by ancilla measurement probabilistically simulates each fractional query Qθ_x.The construction introduces an ancilla, applies rotations and a controlled-Qx operation, then measures the ancilla in the computational basis.Source: B. Simulating fractional queries with… · p. 5
- The successful measurement branch applies Qθ_x up to a global phase, with success probability ps = 1/v^2 ≥ 1−T/p.The bound uses θ = T/p and sin(θ/2) ≤ T/(2p).Source: B. Simulating fractional queries with… · p. 5
- The construction also requires probabilistic implementation of the conjugated fractional query Q−θ_x.This additional operation is needed by the later error-correction procedure.Source: B. Simulating fractional queries with… · p. 5
- A failed simulation projects the ancilla into |1⟩ and applies an undesired Qπ/2_x operation to the system.These undesired operations are treated as errors in the subsequent correction procedure.Source: B. Simulating fractional queries with… · p. 6
C. Approximating segments of control qubits by low Hamming weight states
The construction replaces each fractional-query segment with a full-query circuit whose control register is concentrated on low Hamming-weight states, enabling truncation to far fewer queries while preserving the segment's action approximately.
C. Approximating segments of control qubits by low Hamming weight states
- A size-m segment is formed after replacing each fractional query with a probabilistic controlled-query simulation, with m chosen so that mθ = 1/4.The segment succeeds with probability at least 3/4 when all ancillas are measured in |0⟩.Source: C. Approximating segments of control… · p. 6
- The control register's Hamming weight represents the number of full queries, and its amplitudes are concentrated on basis states with small Hamming weight.This concentration permits approximation by retaining only states whose Hamming weight is at most k.Source: C. Approximating segments of control… · pp. 6–7
- Rearranging the segment into fixed queries with controlled driving unitaries allows truncation based on the retained Hamming weight rather than the original segment size m.The rearranged circuit exactly simulates the original before truncation.Source: C. Approximating segments of control… · p. 7
- The truncated circuit uses k + 1 = O(log(T/ε2ε3) / log log(T/ε2ε3)) full queries and approximates each segment within O(ε2ε3/T).This follows because full queries square to the identity, allowing the final m − k queries to be removed on states of Hamming weight at most k.Source: C. Approximating segments of control… · p. 7
D. Correcting erroneous fractional queries
The error-correction procedure identifies failed fractional queries, reverses and redoes affected segments recursively, and controls the expected overhead so the full simulation remains near-linear in T.
D. Correcting erroneous fractional queries
- Each size-m segment succeeds with probability at least 3/4, while an unsuccessful segment has expected error count at most 1/4.The segment size is selected so that mθ = 1/4, and there are 4T such segments.Source: D. Correcting erroneous fractional… · pp. 7–8
- Measured ancillas reveal the exact error positions, enabling the failed segment to be undone and then redone with Q_x^±π/2 corrections inserted at those positions.Undo and redo steps can themselves fail, so recovery is applied recursively.Source: D. Correcting erroneous fractional… · p. 8
- The recursive recovery process is a biased random walk whose expected number of segment computations is at most 2 because each computation succeeds with probability at least 3/4.A success advances the process, while a failure requires undoing and redoing the failed computation.Source: D. Correcting erroneous fractional… · pp. 8–9
- Undo segments become more expensive with additional errors, but truncation bounds each error's added cost by O(log(T/ε2ε3) / log log(T/ε2ε3)) full queries.The bound applies to undo, recursive undo, and redo operations.Source: D. Correcting erroneous fractional… · p. 9
- The expected number of segment computations and exact correction operations is O(T), with the expected correction count satisfying C0 ≤ 1.This controls the cumulative cost before applying the truncation approximation.Source: D. Correcting erroneous fractional… · p. 10
- With truncation, the expected full-query cost is O(T log(T/ε2ε3) / log log(T/ε2ε3)), and the final state has fidelity at least 1 − 3(ε1 + ε2 + ε3).A cutoff yields termination probability at least 1 − ε2, completing the theorem's simulation guarantee.Source: D. Correcting erroneous fractional… · p. 10
E. Rigorous analysis of fidelity
The fidelity analysis compares ideal and approximated control states through successive output-state relations, showing that the final simulated state remains close to the continuous-time algorithm’s output.
E. Rigorous analysis of fidelity
- The analysis begins with overlap at least √(1 − ε1) between the continuous-time algorithm’s output and the fractional-query algorithm’s output.Source: E. Rigorous analysis of fidelity · p. 10
- The error-correction construction is represented as a unitary whose input includes many control qubits in the exact state |χ⟩⊗O(T/ε2).Source: E. Rigorous analysis of fidelity · p. 10
- The construction augments the simulated output with success information and a fixed auxiliary state before comparing it with the ideal output.Source: E. Rigorous analysis of fidelity · p. 10
- Changing the control input from |χ⟩⊗O(T/ε2) to |χ′⟩⊗O(T/ε2) produces the perturbed output state analyzed in the fidelity bound.Source: E. Rigorous analysis of fidelity · p. 10
- The final fidelity is at least 1 − ε when ε1 = ε2 = ε3 = ε/9.The comparison uses the output state of the continuous-time algorithm and the corresponding portion of the error-corrected simulated state.Source: E. Rigorous analysis of fidelity · p. 10
III. CONCLUDING REMARKS
The paper establishes an efficient discrete-query simulation of continuous-time algorithms and transfers discrete query lower bounds to the continuous-time model, while identifying unresolved overheads.
III. CONCLUDING REMARKS
- O(T log T / log log T) full queries and O(T^2 log T) known unitaries simulate total evolution time T with constant-bounded success probability.The query bound holds for arbitrary precision, while the authors expect but do not provide a reduction in known-operation complexity.Source: §III · p. 11
- Discrete quantum query lower bounds transfer to continuous-time query complexity with at most a sub-logarithmic-factor loss.The polynomial method can transfer its lower bound without the log T / log log T loss, although continuous-time algorithms might still attain that bound when discrete algorithms cannot.Source: §III · p. 11
- The factor log T / log log T arises from segmenting the simulation so each block has small error; removing it requires new error-correction handling.Source: §III · p. 11
- For arbitrary initial states, the approach requires an ε^-1 query-complexity factor for failure probability ε, although the authors conjecture improvement toward O(log(1/ε)).Source: §III · p. 11
- For time-independent driving Hamiltonians, many operations can be reduced to O(T polylog T), but the unresolved R2 gates prevent establishing that total bound.Source: §III · p. 11