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Quantum walk speedup of backtracking algorithms

Ashley Montanaro

arXiv:1509.02374v2quant-phcs.DS

TL;DR

Classical backtracking can solve structured CSPs more efficiently than unstructured Grover search, but its tree may still be exponentially large. The paper uses a Belovs-based quantum walk on the implicitly defined backtracking tree to obtain near-quadratic speedups, with suitable input distributions yielding exponential reductions in expected runtime. These guarantees apply broadly, including DPLL, while their average-case amplification has important limitations.

  • Problem

    Generic Grover search ignores structure that classical backtracking exploits, motivating a quantum method that accelerates backtracking-based CSP solving.

    Source: §1 · p. 1

  • Method

    A discrete-time quantum walk searches for marked partial solutions in the implicitly generated backtracking tree, using predicate P and heuristic h.

    Source: §1.2 · p. 3, §1.1 · p. 3

  • Results

    O(√T n^(3/2) log n) evaluations of P and h solve the search-or-no-solution task with bounded error, and the framework applies to DPLL and other backtracking algorithms.

    Source: §1.1 · p. 3

  • Takeaways & Limitations

    The approximately quadratic per-instance speedup can become an exponential separation in expected runtime for suitable input distributions.

    Source: §4 · p. 13

    Average-case exponential separation is uncertain because heavy tails may be finite-size effects and satisfiable instances may terminate early while the quantum algorithm uses the whole tree.

    Source: §4 · p. 14

Abstract

from arXiv · show

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.

1 Introduction

The paper develops a quantum-walk method that accelerates classical backtracking for CSPs, including DPLL, with near-quadratic instance-dependent speedups and possible exponential average-case reductions.

1.1 Results

  • O(√T n^(3/2) log n) oracle evaluations suffice to find a CSP solution or report none exists, versus a classical backtracking tree of T vertices.The algorithm has bounded error, polynomial space, and can be modified for unique-solution instances with a different logarithmic factor.Source: §1.1 · p. 3
  • The method applies to arbitrary backtracking predicates and heuristics, including randomized heuristics and DPLL with the unit-clause heuristic.Its instance-dependent bound does not require knowing T in advance, and quick classical instances remain quick quantum instances.Source: §1.1 · p. 3
  • The speedup is approximately quadratic per instance and can become an exponential separation in expected runtime under suitable input distributions.The amplification concerns average runtimes rather than the per-instance quadratic relationship.Source: §1.1 · p. 3

1.2 Techniques

  • The quantum algorithm represents partial assignments as vertices of a backtracking tree and uses a discrete-time quantum walk to search for marked vertices.The tree is defined implicitly by the predicate P and branching heuristic h, and the walk starts at the root rather than from a known stationary distribution.Source: §1.2 · pp. 3–4
  • The main quantum subroutine specializes Belovs’ effective-resistance walk result to trees and extends detection to finding marked vertices.Binary search finds a marked vertex, while eigenvector analysis gives a faster method when the marked vertex is unique.Source: §1.2 · p. 4

1.3 Other prior work

  • Unlike prior approaches tied to known tree structure, restricted predicates, leaf mappings, or randomized subroutines, these bounds apply to arbitrary classical backtracking algorithms.The paper contrasts its framework with nested Grover search, continuous-time walks, leaf-based search, certificate search, and amplitude amplification.Source: §1.3 · pp. 4–5

1.4 Organisation

  • The paper organizes its development around tree-walk detection, finding extensions, CSP backtracking applications, average-case runtime reductions, and improvement barriers.The preliminaries introduce effective spectral gaps, phase estimation, and measurement-distance facts used in the quantum analysis.Source: §1.4 · p. 5, §1.5 · p. 6

2 Quantum walks on trees

The paper develops a quantum walk on rooted trees for detecting and locating marked vertices, then extends it to search efficiently under bounded-degree and unique-marked-vertex conditions.

2 Quantum walks on trees

  • Quantum walk detection uses local diffusion operators on a rooted tree and alternates the reflections RA and RB.Marked vertices use identity diffusion, while unmarked vertices use inversions about locally defined states; one walk step is RBRA.Source: §2 · p. 7
  • O(sqrt(Tn) log(1/δ)) applications of RA and RB detect whether a marked vertex exists with failure probability at most δ.The procedure applies phase estimation repeatedly and thresholds the number of eigenvalue-1 outcomes.Source: §2 · pp. 7–8

2.1 Finding a marked vertex

  • O(n) subtree checks locate a marked vertex, yielding runtime O(sqrt(T) n^(3/2) log n log(1/δ)) under bounded degree.Unknown tree size is handled by doubling guesses for T, while verification of the returned vertex preserves correctness.Source: §2.1 · pp. 8–9
  • With k marked vertices, repeating the search finds all of them in O(k sqrt(T) n^(3/2) log n log(k/δ)) time.Previously found marked elements are excluded from subsequent searches.Source: §2.1 · p. 9

2.2 Search with a unique marked element

  • A unique marked vertex improves the search bound by almost a factor of n and is never worse than classical O(T) up to logarithmic factors.The improvement is most relevant for tall, thin trees.Source: §2.2 · p. 9
  • The unique-marked-vertex analysis assumes the marked vertex has known depth after an O(log n)-repetition depth search, with failure probability reduced by repetition.The paper notes further barriers to improving the complexity and applicability of these algorithms.Source: §2.2 · pp. 9, 11
  • Measuring a state that encodes the root-to-marked-vertex path yields a uniformly distributed intermediate vertex, reducing the expected number of search stages to logarithmic in n.The algorithm recursively searches the subtree rooted at the measured vertex; state preparation is approximated by phase estimation.Source: §2.2 · pp. 9–11

3 From quantum walks on trees to accelerating backtracking

The paper maps a classical backtracking computation to a rooted tree of valid partial assignments and implements the quantum walk reflections using the predicate P and heuristic h.

3 From quantum walks on trees to accelerating backtracking

  • The construction assumes efficient evaluation of P and h and bounded local domain size, while tree reweighting is noted as a possible route to improving polylogarithmic factors.The reweighting idea is attributed to Alexander Belov in a footnote.Source: §3 · p. 11
  • Each tree vertex encodes a valid partial assignment, connected to extensions chosen by the backtracking heuristic h that are not rejected by P.The root is the empty assignment, and each nonroot vertex connects to its parent and valid child extensions.Source: §3 · p. 11
  • The quantum algorithm represents assignments in a Hilbert space whose registers store the level and index-value pairs, using O(log n+log d) qubits per subsystem.Ancilla registers hold temporary values used to compute child information and implement reflections.Source: §3 · pp. 11–12
  • The procedure returns when P(x) is true; otherwise, for odd ℓ it subtracts h(...) from iℓ and swaps a with vℓ, adjusts ℓ when a ≠ ∗, forms S, and applies the specified reflection.Source: §3 · p. 12
  • The reflection states are implemented through unitaries Uα,S, with P and h treated as black-box oracles and constant-time oracle overhead under the stated assumptions.Approximate universal-gate implementations add at most a polylogarithmic runtime factor.Source: §3 · p. 12
  • The resulting operators RA and RB are correct on basis states and superpositions, using O(1) auxiliary operations per operator use.This completes the implementation needed to apply the quantum-walk algorithm to the backtracking tree.Source: §3 · pp. 12–13

4 From quadratic speedups to exponential speedups

The section shows that input distributions can amplify instance-wise quadratic quantum backtracking speedups into exponential average-runtime separations, while identifying important limitations and a 3-SAT construction.

4 From quadratic speedups to exponential speedups

  • Power-law runtime tails can yield exponential separation, but finite-size effects, early satisfiable solutions, and unsatisfiable-instance behavior make the phenomenon uncertain.The quantum algorithm depends on the whole backtracking tree, unlike a classical algorithm that may terminate after finding a solution early.Source: §4 · p. 14
  • Random branching choices on a single fixed instance do not provide the same separation, because bounded quantum expectation implies a classical cutoff succeeds with constant probability.Markov’s inequality gives T(r) = O(R^2) with 99% probability when the quantum expected runtime is at most R.Source: §4 · p. 15
  • Exponential average-runtime separations arise when quantum backtracking is combined with suitable heavy-tailed distributions of classical tree sizes.The section explains that distributions over constraint densities can produce this effect more naturally than contrived hard-versus-trivial mixtures.Source: §4 · pp. 13, 15

4.1 Expected runtime bounds

  • Na¨ıveBt has rigorously bounded expected backtracking-tree size on random k-SAT, with k = 3 giving C ≤ 0.907/√α and C′ ≥ 0.906/√α − 0.142/α^2.The bounds apply for random k-SAT with m = αn clauses in the stated density range.Source: §4.1 · p. 15
  • For distributions supported on sufficiently dense 3-SAT instances, the probability of satisfiability is O(2^-n), allowing the analysis to ignore the extra cost of finding all solutions.The stated sufficient condition is m > 16n/(ln 2).Source: §4.1 · pp. 15–16
  • For 3-SAT, averaging over clause counts under pm ∝2^(-0.454n^(3/2)/√m) yields an average-case exponential quantum-classical separation.The construction varies the number of clauses while keeping the number of variables fixed, though the authors call the distribution somewhat unnatural.Source: §4.1 · p. 16

5 Improving the quantum walk algorithm?

The section identifies three main obstacles to improving quantum tree search: depth dependence, multiple marked vertices, and the cost of finding all marked vertices.

5 Improving the quantum walk algorithm?

  • Tree-search depth cannot generally be removed: some depth-O(n) trees require Ω(√(Tn)) queries even when the tree is known.A path already requires Ω(T) steps when the marked vertex is at its end.Source: §5 · p. 16
  • Finding all k marked vertices may require more than the ideal unstructured-search scaling, and existing quantum-walk results do not establish the needed general bound.Known approaches can incur extended hitting time and assume a known graph with a stationary-distribution initial state, unlike this setting.Source: §5 · pp. 16–17
  • An algorithm finding one of multiple marked vertices in o(√(Tn)) queries would contradict the lower bound for finding a unique marked leaf.Attaching a marked depth-O(log k) subtree would let the algorithm recover the original leaf without additional queries.Source: §5 · p. 17
  • When classical backtracking is lucky enough to find a solution early, quantum exploration of the whole tree may not outperform it.The authors suggest biasing the quantum walk toward regions visited earlier by the classical algorithm as a possible improvement.Source: §5 · p. 17

A Proof of technical claim for search with one marked element

The appendix proves the technical spectral-gap claim underlying quantum search for a single marked vertex by constructing a vector satisfying the two projector constraints.

A Proof of technical claim for search with one marked element

  • The proof constructs |ξ⟩ so that ΠA|ξ⟩ = 0 and ΠB|ξ⟩ = |φ⊥⟩, enabling the effective spectral gap lemma.The construction uses invariant-subspace vectors supported on vertices and their children.Source: A Proof of technical claim for search… · p. 18
  • Coefficients are propagated along the path to the unique marked vertex and constrained to match across off-path parent-child relations.Separate cases handle whether the marked vertex lies in A or B, with αx0 = 0 in the latter construction.Source: A Proof of technical claim for search… · pp. 18–19
  • The resulting coefficient bounds imply ∥Pχ|φ⊥⟩∥ = O(χ√(Tn)), which supplies the required spectral estimate.The bound follows by applying the effective spectral gap lemma to the constructed vector.Source: A Proof of technical claim for search… · p. 19

B The runtime of the Na¨ıveBt backtracking algorithm for k-SAT

The appendix analyzes the Na¨ıveBt backtracking tree on random k-SAT by estimating partial-assignment consistency and maximizing the resulting expected vertex count.

B The runtime of the Na¨ıveBt backtracking algorithm for k-SAT

  • For random k-SAT with m = αn clauses, Na¨ıveBt branches on the lowest-index unassigned variable and yields explicit expected-tree-size bounds.The analysis covers 1 ≤ α ≤ n^(k−1), with constants depending only on α and k.Source: B The runtime of the Na¨ıveBt… · p. 19
  • The derivation estimates the probability that each partial assignment is clause-consistent, sums over assignments, and maximizes the resulting expression over the assigned-prefix length.Derivative and rounding corrections affect the final bound only by an O(1) factor.Source: B The runtime of the Na¨ıveBt… · pp. 19–20
  • When α is much larger than 2^k/ln 2, the probability that a random instance is satisfiable is exponentially small in n.This supports focusing on unsatisfiable instances in dense regimes.Source: B The runtime of the Na¨ıveBt… · p. 19
  • For k = 3, the constants satisfy C ≤ 0.907/√α and C′ ≥ 0.906/√α − 0.142/α^2.These bounds quantify the exponential rate governing the expected number of vertices.Source: B The runtime of the Na¨ıveBt… · p. 20
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