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A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits

Matthew Amy, Dmitri Maslov, Michele Mosca, Martin Roetteler

arXiv:1206.0758v3quant-phcs.ET

TL;DR

Efficient exact synthesis of quantum circuits is needed for fixed fault-tolerant instruction sets, but brute-force search is exponentially costly. The paper uses meet-in-the-middle search to optimize circuit depth and related costs, reporting faster synthesis and improved circuits while remaining practical only for small instances.

  • Problem

    Fixed fault-tolerant instruction sets require efficient decompositions of logical operations, while brute-force synthesis has exponential complexity.

  • Method

    The paper uses a meet-in-the-middle algorithm to search for depth-optimal circuits and adapt the search to other cost functions, including sequential non-Clifford gates.

  • Results

    The implementation produced an exact minimal-depth controlled-H decomposition in under 0.500s, whereas a four-level Solovay-Kitaev implementation took over 2 minutes and produced over 1000 approximate gates.

  • Takeaways & Limitations

    The algorithm finds minimal-depth circuits and reports circuits better than those found in the literature for small quantum-circuit synthesis problems.

  • Takeaways & Limitations

    Because the runtime remains exponential and full unitary matrices cannot be stored for sufficiently deep searches, the method is limited to small circuits and requires a space-time trade-off.

Abstract

from arXiv · show

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

1 Introduction

Quantum circuit synthesis must translate high-level logical operations into efficient circuits over fixed fault-tolerant gate sets. This paper addresses exact synthesis with an algorithm intended to automate optimization as quantum devices grow more complex.

  • Motivation: Fixed instruction sets make efficient implementation of high-level quantum gates essential because fault-tolerance protocols constrain available operations.Quantum algorithms use varied gates, while hardware instruction sets contain relatively few fault-tolerant logical gates.
  • Motivation: Growing quantum devices make hand optimization less feasible, increasing the need for automated circuit-design tools.The paper links this need to advances in quantum information processing and fault-tolerant thresholds.
  • Contribution: A roughly square-root speedup over brute force is obtained for computing optimal circuits, including optimization beyond depth.The method can optimize weighted gate costs and sequential non-Clifford gates, while remaining exponential overall.
  • Prior work and contribution: The approach builds on search techniques from reversible-circuit synthesis and differs from prior restricted models by targeting general unitary circuits.Earlier methods include canonical forms for single-qubit circuits and formal-logic approaches restricted to Boolean controls.
  • Prior work and contribution: The paper presents exact decompositions for logical gates, while noting that the algorithm can also be extended to approximation sequences.Prior work emphasized approximations in small state spaces; this work focuses on exact decompositions.

2 Preliminaries

The paper models quantum circuits as compositions of parallel depth-one layers over an instruction set and defines depth through critical paths. Its main result decides bounded-depth implementability with a search whose complexity improves on brute force, while remaining exponential.

  • Quantum circuit model: Quantum gates are unitary operators on the n-qubit state space, and circuits compose these operators sequentially or in parallel.Gates act non-trivially on selected qubits and as the identity on the remainder.
  • Circuit depth: Circuit depth is the length of a longest critical path through the directed acyclic graph of gates.This definition captures the parallel execution structure of circuits.
  • Circuit representation: An instruction set is a fixed gate set containing each gate’s inverse, and V_n,G denotes the unitaries realizable by depth-one circuits over that set.A depth-at-most-m circuit is represented as a product of m elements of V_n,G.
  • Main result: The algorithm determines whether a target unitary can be implemented over G at depth at most l and can return a minimal-depth circuit when one exists within that bound.The search is based on splitting the circuit depth and matching circuit halves.
  • Complexity and scope: The search improves on brute-force complexity O(|V_n,G|^l), with practical runtime close to Θ(|V_n,G|^⌈l/2⌉), but remains exponential and is practical only for small qubit counts.The exponential dependence follows from the growth of |V_n,G| with the number of qubits.
  • Instruction sets: The instruction set {H, P, P†, CNOT, T, T†} is universal because {H, P, CNOT} generates the Clifford group and adding a non-Clifford gate yields density.The paper motivates this set through fault-tolerant quantum computation.

3 Search Algorithm

The algorithm finds minimum-depth exact circuits by splitting candidate circuits into two halves and detecting meet-in-the-middle collisions, while supporting alternative cost criteria and ancilla-assisted searches. Its practical scope is limited by exponentially growing search spaces, especially Clifford groups for larger qubit counts.

  • Search Algorithm: The main insight is that circuits of depth l can be searched by generating circuits only up to depth ⌈l/2⌉.This reduces the generation depth relative to brute-force enumeration while preserving exact depth-optimal search.
  • Search Algorithm: Lemma 1 characterizes a depth-l implementation of U through a collision between inverse partial circuits and forward partial circuits.A depth-l circuit is divided into portions of depths ⌊l/2⌋ and ⌈l/2⌉, with V W = U and W = V†U.
  • Search Algorithm: The algorithm generates circuit sets of increasing depth and searches for intersections S†_i U ∩ S_i, returning a circuit of minimum depth when a collision is found.For a solution within the limit, termination occurs at the smallest depth supporting an implementation.
  • Search Algorithm: The meet-in-the-middle search uses logarithmic-time lookup after sorting circuit sets under a strict lexicographic ordering.The implementation searches candidate products against stored sets rather than comparing every pair directly.
  • Optimizing different cost functions: The method can optimize weighted gate cost and T-depth, including searches that alternate Clifford and non-Clifford phases.Strictly positive gate weights make minimum cost increase with depth; for Clifford+T circuits, T-depth is searched using generated Clifford groups and T phases.
  • Limitations: For 3 qubits, the Clifford group contains 92,897,280 elements up to global phase, making higher-dimensional searches impractical with this approach.The authors note that C4 would not fit in a computer with reasonable memory using this method.
  • Ancilla-assisted synthesis: Ancilla-assisted synthesis is supported for ancillas initialized and returned to |0⟩, but searching only relevant input columns requires generating and searching more circuits.The restricted search compares the first 2^n columns while preserving the desired action on inputs of the form |0⟩^⊗m|ψ⟩.

4 Search tree pruning

Search-tree pruning stores one minimal-depth circuit for each canonical equivalence class, identifying circuits related by qubit relabeling, inversion, or global phase. Canonicalization reduces redundant exploration while preserving minimum depth within each class.

  • Search tree pruning: The search tree is generated breadth first, with each set S_i representing one depth level and each branch selecting the next gate.This organization supports pruning newly generated circuits against previously stored representatives.
  • Equivalence classes: The equivalence relation groups unitaries differing by qubit relabeling, inversion, or global phase, and stores only one minimal-depth representative per class.The pruning procedure computes a canonical representative and checks whether its class has already been encountered.
  • Canonical representatives: Canonical representatives are chosen lexicographically after generating qubit permutations and inverses, with an added O(n!) overhead.For an n-qubit unitary, the method considers 2^n n! permutations and inversions before retaining the minimum.
  • Global phase: When the gate ring permits it, eight possible global phase factors are combined with permutations and inversions before selecting the lexicographically earliest representative.This phase-aware canonicalization applies to the specified ring representation.
  • Global phase: Canonical phase selection uses a reference matrix element to remove phase equivalences efficiently and enables symbolic comparisons over the underlying ring.The reference is the first nonzero matrix element encountered row by row.
  • Equivalence classes: Permutations and inverses preserve circuit depth, so every unitary in an equivalence class has the same minimum circuit depth.Inverse circuits are obtained by reversing gates and replacing each with its inverse.

5 Implementation Details

The implementation uses ordered data structures and compact circuit representations to make meet-in-the-middle searches practical, while trading storage space against recomputation. Full unitary storage remains infeasible for sufficiently deep searches, especially as circuit size grows.

  • Search structures: Red-black trees reduce collision searches from O(|S_i||S_j|) comparisons to logarithmic search in the ordered database.The implementation imposes a lexicographic ordering on generated circuits and stores each S_i as a red-black tree.
  • Search structures: At most 52 of 1,316,882 distinct 3-qubit unitaries mapped to any one hash value, but hash tables showed no performance improvement.The code used a libstd++ hash-table implementation despite reasonable collision rates.
  • Memory limits: Full unitary matrices cannot be stored for searches to significant depth, so the implementation must make a space-time trade-off.Compression and equivalence-class representatives reduce storage, but do not remove the underlying scalability boundary.
  • Memory limits: More than 10^12 circuits already occur at depth 5 for 3 qubits under the standard universal instruction set, making exact unitary storage prohibitive.Each exact 3-qubit unitary requires 5 × 64 integers, so all depth-5 circuits would require more than 1 petabyte; experiments found at most 36,042,958 equivalence classes.
  • Representation trade-offs: Storing circuits as depth-1 gate lists reduces storage, while matrix keys accelerate searches without recomputing each circuit's unitary.Each depth-1 circuit uses n bytes, and an m × m matrix key is stored with each circuit; m = 1 sufficed for interesting depths up to 4 qubits.
  • Representation trade-offs: Floating-point-generated keys require symbolic computations and direct key computation from unitaries so equal unitaries receive equal numerical error.Attempts to avoid floating-point computations produced too many key collisions for practical use, and better key-generation methods remained under investigation.

6 Performance and Results

The meet-in-the-middle implementation produced depth-optimal decompositions for several logical gates and substantially improved selected Clifford+T circuits. Searches and database generation remained exponentially challenging as circuit size and qubit count increased.

  • Performance: The meet-in-the-middle implementation followed a compressed exponential growth curve, and increasing qubit counts increased search difficulty because of matrix-multiplication costs.The implementation was tested on a server with 16 GB RAM plus 16 GB swap space.
  • Depth-optimal implementations: Under 0.500s, the algorithm generated an exact minimal-depth controlled-H decomposition, whereas Solovay–Kitaev took over 2 minutes for an approximation exceeding 1000 gates.The approximation had error 0.340 in Fowler’s distance metric.
  • Depth-optimal implementations: The Toffoli decomposition reduced total depth from 12 to 8 while preserving a depth-optimal implementation found by the search.The reported circuit is the controlled-CNOT among the three-qubit logical gates examined.
  • Depth-optimal implementations: The controlled-T optimization reduced T gates from 15 to 9, CNOT gates from 16 to 12, and T-depth from 9 to 5.The same peephole approach reduced the one-bit adder’s T gates from 14 to 8 and T-depth from 8 to 2.
  • Optimal T-depth implementations: A T-depth 3 Toffoli circuit was found, reducing the prior 5 T-stages to 3 and providing an approximate 40% speed-up under negligible Clifford-gate cost.The authors conjecture T-depth 3 is minimal because Toffoli appears to require at least 7 T gates, but minimality with zero ancillas is unproved.

7 Conclusion

The paper presents a minimal-depth quantum-circuit synthesis algorithm and reports improved circuits for important logical gates. Its practical scope is limited by exponential complexity, motivating approximation methods and database-based optimization for larger circuits.

  • Conclusion: The algorithm finds minimal-depth circuits for a specified unitary and gate set, while also supporting minimization of sequential non-Clifford gates.The implementation reports better circuits than those found in the literature.
  • Computational efficiency: 32 minutes generates all equivalence-class representatives of 3-qubit circuits up to depth 4, while 415 seconds finds an optimal 3-qubit circuit up to depth 8.
  • Computational efficiency: Matrix-based representations make 4-qubit unitaries memory-intensive and make circuit composition, permutation, and inversion computationally expensive.A 4-qubit unitary may require 40,960 bits without compression, 640 times the reversible-function representation.
  • Computational efficiency: Storing circuits instead of entire unitaries reduces the minimum storage from 4^n × 5 × 4 bytes to as few as n bytes, at a space-time tradeoff.Computing a circuit of depth d still requires d matrix multiplications.
  • Limitations and future work: Future work targets approximate synthesis because many operations, including phase rotations e2πi/2^m for m ≥ 4, cannot be implemented exactly with Clifford and T gates.The paper suggests meet-in-the-middle techniques could accelerate searches for approximate sequences.
  • Limitations and future work: The method remains exponential in instruction-set size and depth, making it unlikely to be useful for large circuits directly.The authors propose using generated circuit databases for peephole optimization and circuit re-synthesis.
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