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A Depth-Optimal Canonical Form for Single-qubit Quantum Circuits

Alex Bocharov, Krysta M. Svore

arXiv:1206.3223v1quant-ph

TL;DR

Efficient fault-tolerant decomposition of arbitrary single-qubit operations requires compact exact or approximate representations with low resource costs. The paper develops a depth-optimal canonical form over {H, T}, associated reduction and database-search algorithms, and a polylogarithmic Solovay–Kitaev procedure. Its methods produce substantially improved approximation precision and T-counts while using less classical memory than prior implementations.

  • Problem

    Arbitrary single-qubit operations must be decomposed into discrete fault-tolerant gates, but approximation and recursive synthesis make circuit depth, T-count, and computational resources important optimization targets.

  • Method

    The paper uses a PSU(2)-based canonical form over {H, T}, reduction rules, a searchable canonical-circuit database, and an extension to Solovay–Kitaev decomposition.

  • Results

    The approach finds significantly improved approximation precision and T-count/precision trade-offs, including ϵ = 5 × 10^-8 below 500 T gates and 120 versus roughly 100,000 T gates at ϵ = 5×10^-5.

  • Takeaways & Limitations

    Canonical representations can reduce the resources required for single-qubit decomposition and support efficient Solovay–Kitaev implementations.

Abstract

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Given an arbitrary single-qubit operation, an important task is to efficiently decompose this operation into an (exact or approximate) sequence of fault-tolerant quantum operations. We derive a depth-optimal canonical form for single-qubit quantum circuits, and the corresponding rules for exactly reducing an arbitrary single-qubit circuit to this canonical form. We focus on the single-qubit universal H,T basis due to its role in fault-tolerant quantum computing, and show how our formalism might be extended to other universal bases. We then extend our canonical representation to the family of Solovay-Kitaev decomposition algorithms, in order to find an \epsilon-approximation to the single-qubit circuit in polylogarithmic time. For a given single-qubit operation, we find significantly lower-depth \epsilon-approximation circuits than previous state-of-the-art implementations. In addition, the implementation of our algorithm requires significantly fewer resources, in terms of computation memory, than previous approaches.

I. INTRODUCTION

The paper targets efficient fault-tolerant decomposition of arbitrary single-qubit operations, emphasizing the {H, T} basis, depth, and non-Clifford cost. It introduces a canonical representation and algorithms for exact or approximate decomposition, database search, and polylogarithmic Solovay–Kitaev approximation.

  • Optimization objective: The method simultaneously minimizes circuit depth and T-count, motivated by the higher fault-tolerant cost of non-Clifford T gates.T-count is the number of non-Clifford T gates in a sequence.
  • Relation to prior work: Relative to prior approaches, the framework improves exact-search complexity for {H, T} circuits and avoids costly uniqueness checks used by earlier canonical-search methods.For single-qubit exact decomposition, the stated complexity improves from O(d |B|^d/2) to O(d |B|^d/4), with B = {H, T}.
  • Canonical representation: The paper introduces a single-qubit canonical form and reduction rules for converting circuits into that form.The representation uses PSU(2) and expresses circuits as g1.c.g2, where g1 and g2 are Clifford gates and c is uniquely canonical.
  • Decomposition algorithms: The authors develop an algorithm for exact depth-optimal decomposition when possible, or depth-optimal ϵ-approximation otherwise.This addresses both exact synthesis and approximation within a unified decomposition objective.
  • Database search: They develop a compact canonical-circuit database and efficient lookup procedure for searching candidate decompositions.The database supports practical applications including Solovay–Kitaev decomposition.
  • Solovay–Kitaev application: The canonical representation is extended to Solovay–Kitaev algorithms to find single-qubit ϵ-approximations in polylogarithmic time.This addresses the sensitivity of recursive Dawson–Nielsen decompositions to the cost and precision of their base circuits.

II. A CANONICAL FORM AND CANONICAL REDUCTION OF CIRCUITS

The paper represents single-qubit {H,T} circuits canonically by factoring out Clifford gates and reducing circuits over the equivalent {TH, SH} basis. Canonical circuits uniquely represent Clifford double cosets, while normalization and reduction rules constrain their structure and relate T-count to depth.

  • Canonical representation: Every {H,T} circuit can be represented as g1.c.g2, where c is canonical and g1,g2 are Clifford gates from the 24-element subgroup C.The representation follows from first obtaining a normalized form and then identifying the canonical representative of the circuit’s Clifford double coset.
  • Canonical basis: The {H,T} basis is recast using syllables TH and SH, with SH having order 3 and TH having infinite order in PSU(2).The identities relating {TH, SH} and {H,T} establish an equivalent universal single-qubit basis.
  • Circuit reduction: Reduction rules eliminate interior (SH)^2 patterns, yielding normalized circuits that end in TH and contain no explicit (SH)^2.The pattern TH(SH)^2TH collapses to H while removing two TH syllables.
  • Canonical circuits: A normalized circuit is canonical when it contains no SH before the fifth syllable, so the shortest canonical circuit containing SH is (TH)^4SH.TH.There are only six canonical circuits with fewer than six syllables: I, TH, (TH)^2, (TH)^3, (TH)^4, and (TH)^5.
  • Uniqueness and cost: C-equivalent canonical circuits are syntactically identical, making the canonical representative unique within each Clifford double coset.This uniqueness makes T-count an invariant of the gate represented by a canonical circuit.
  • Uniqueness and cost: In the {H,T} canonical form, T-count and overall depth are closely tied: the representation has between T-count − 1 and T-count + 1 Clifford gates.All but at most two Clifford gates are H or HSH, and the number of HSH sequences is less than T-count − 3.

III. DEPTH-OPTIMAL CIRCUIT DECOMPOSITION

The canonical representation supports exact and approximate depth-optimal decomposition by reducing search to canonical circuits and Clifford cosets. It lowers representation and search costs while preserving efficient normalization and canonicalization procedures.

  • Approximate decomposition: Canonical circuits replace database searches over all unique gates with searches over Clifford double cosets, whose optimal ε-approximation is inherited from a canonical representative.At most 576 coset elements need consideration, and the searches can be performed in parallel.
  • Reduction complexity: Normalization of an {H,T} circuit costs linear time in its size, while canonicalization costs quadratic time in the normalized circuit’s T-count.The canonicalization procedure uses commutation relations between Clifford elements and T gates.
  • Reduction complexity: The inverse of a normalized circuit has a structured normalized representation of the form H.c′.H or H.c′.H.S^3, with c′ computable in time linear in the original depth.The inverse itself need not be normalized.
  • Resource requirements: The canonical database has 2^(t−3) + 4 circuits with T-count at most t; at t = 24, it contains 2,097,156 circuits and uses approximately 900 MB.The reduced storage supports maintaining substantially longer canonical circuits in practice than recursive databases.

IV. SEARCH FOR CANONICAL APPROXIMATIONS

The search algorithm organizes canonical circuits by absolute trace and geometric indices on Clifford-partitioned trace levels. Empirical scaling observations support efficient ε-approximation queries by narrowing searches to relevant trace levels and local geometric regions.

  • Query reduction: The database query seeks the minimum-cost ε-approximation of a target gate, and Clifford reduction allows the search to use canonical circuits without increasing cost.The framework applies to any basis with Clifford reduction, although the section focuses on {H,T} with T-count as cost.
  • Trace indexing: Absolute matrix trace is the database’s primary key because ε-closeness bounds trace differences by ||tr(U)|−|tr(V)|| < 4ε.The method assumes the specified trace-distance measure, though other distance measures are possible.
  • Geometric indexing: The search distributes approximation targets across nonintersecting trace-indexed node groups, allowing most secondary indices to be eliminated using trace-level geometry.Targets near edges or vertices require multiple tile, edge, or vertex indices.
  • Empirical scaling: Canonical circuits with T-count ≤ k have empirically at most 6 × 2^(k/2) distinct absolute trace values, while each trace level contains at most O(2^(k/2)) circuits of T-count k.The second bound is stated conditionally on a conjecture for constancy of T-count within a trace level.
  • Empirical scaling: Search over canonical circuits with T-count ≤ k costs O(ε^k 2^k) when an approximation exists, while non-existence is detected in O(k) average steps and O(k 2^(k/2)) worst-case steps.These estimates are empirical observations for the database search.
  • Geometric indexing: Trace levels are partitioned into 24 Clifford-related tiles, with edge and vertex indices added to handle targets near tile boundaries.The geometry includes 14 special vertices where more than two tiles meet; interior targets usually require only the relevant face index.

V. APPLICATION TO SOLOVAY-KITAEV DECOMPOSITION

The canonical representation is integrated into Dawson–Nielsen Solovay–Kitaev decomposition to improve approximation quality and reduce circuit cost through lookup and canonical reduction.

  • Solovay–Kitaev recursion: Improving 0-level precision by expanding the database from T-count ≤12 to ≤28 improves mean precision 9.8-fold and can exponentially reduce later circuit T-count.The D-N estimate predicts precision coefficients around 10^-6 at depth 4 and 10^-9 at depth 6.
  • Solovay–Kitaev recursion: The method can exceed the D-N estimate: with 0-level databases up to T-count 25 or 26, accuracy becomes machine-double limited by recursion depth 4.Experiments therefore cover recursion depths ≤3, while the longest approximations barely exceed T-count 3000.
  • Canonical reduction: Canonical reduction simplifies normalized compositions through cancellations, reducing the resulting circuit’s T-count in many cases.The TH.(SH)^2.TH pattern reduces to H and can trigger further cancellations.
  • Canonical reduction: Canonical reduction also restricts normalized interior gates to H or HSH, with at most T-count −5 interior HSH gates in canonical circuits.
  • Solovay–Kitaev recursion: Normalizing a D-N approximation yields a minimal-cost circuit exactly equivalent to that circuit, but not necessarily the lowest-cost approximation within the target’s ε-neighborhood.

VI. EXPERIMENTAL RESULTS

Experiments evaluate canonical databases and canonical reduction on 10,000 random single-qubit unitaries, comparing T-count–precision tradeoffs with prior methods.

  • 0-level approximations: Figure 2 compares mean trace-distance precision versus T-count for depth-optimal 0-level approximations from the canonical database and Fowler’s technique.Both methods are evaluated over 10,000 random unitaries.
  • Solovay–Kitaev decomposition: Figure 3 compares mean trace-distance precision versus T-count across recursion levels n = 0, 1, 2, 3 for canonical implementations and D-N.The canonical databases store circuits up to T-count 24, 25, or 26 depending on size.
  • Solovay–Kitaev decomposition: At ε = 5 × 10^-8 below T-count 500, canonical Solovay–Kitaev methods improve precision by a factor of 10^-6 over D-N; at ε = 5 × 10^-5, SK+2G uses 120 T gates versus roughly 100,000 for D-N.The comparison covers recursion levels n = 0, 1, 2, 3 and three database sizes.

VII. CONCLUSIONS AND FUTURE WORK

The paper presents a depth-optimal canonical form and reduction rules for single-qubit circuits, demonstrates gains in approximation tradeoffs, and identifies broader generalizations and a conjectural compression direction.

  • Conclusions: The canonical form and reduction rules improve database size and achieved precision for depth-optimal 0-level approximation, and improve T-count–precision curves for recursive Solovay–Kitaev decomposition.
  • Future work: Future work includes extending canonical forms to multi-qubit gates and other universal bases.
  • Future work: Lossy compression seeks approximately equivalent circuits within distance ε of a target while reducing a cost such as T-count or gate count.The paper states that this direction likely requires Conjecture 1.
  • Conjecture 1: Conjecture 1 asserts that canonical circuits with different T-counts have different absolute trace values.Its empirical brute-force support currently extends only through T-count ≤31.

Appendix A: Elements of the C group

The appendix lists elements of the Clifford group and identities used to rewrite products involving T gates into forms suitable for canonical reduction.

  • Clifford-group elements: The appendix enumerates Clifford-group elements G0 through G23 as H- and S-gate products.
  • Rewrite identities: It gives rewrite identities for products G_i.T, expressing them as equivalent circuits with T repositioned and Clifford factors simplified.
  • Canonical identification: These identities support efficient identification of normalized circuits in the right C-coset and canonical circuits in the corresponding double C-coset.

Appendix C: Proof of Propositions 1 and 3

The proof normalizes arbitrary ⟨H,T⟩ circuits by collecting Clifford and T factors, then repeatedly commuting and combining them to eliminate interior Clifford gates other than H and HSH.

  • Reduction algorithm: The reduction scans a gate list, applies the C/T commutation table, combines adjacent Clifford factors, and recursively processes the remaining list.Clifford factors are merged when adjacent, while T gates are moved through Clifford gates using fixed commutation rules.
  • Complexity: The algorithm eliminates interior Clifford gates other than H and HSH, with constant cost per elimination and linear cost in the input length.This establishes the efficiency of normalization for the gate types targeted by the reduction.

Appendix D: Proof of Propositions 2 and 4

The proof converts normalized circuits into canonical form by pushing SH syllables out of early positions, using rewrite identities and induction on T-count.

  • Proof setup: The proof begins from representations that may start with H or SH and applies identities established by brute-force calculation in PSU(2).This setup reduces the proof to normalized circuits whose relevant suffix starts with TH.
  • Rewrite strategy: The proof transforms prefixes containing early SH syllables into equivalent forms whose Clifford factors surround longer T,H patterns.The listed identities are used to squeeze SH syllables from the first four syllables into surrounding Clifford factors.
  • Inductive cases: If the transformed suffix is empty or starts with H, the resulting circuit is directly put into canonical form; a suffix beginning with TH requires recursive case analysis.The three cases distinguish the starting form of the recursively normalized suffix.
  • Complexity: The potentially quadratic reduction cost arises only from the final case, where a normalized subcircuit must be recursively brought back to canonical form.All other subcircuit normalizations are linear, and the quadratic pattern was not observed in the reported experiments.

Appendix E: Proof of Theorem 1

The proof of Theorem 1 uses the adjoint action on su(2) to show that normalized circuits cannot represent nontrivial Clifford elements, establishing the uniqueness property needed for canonical reduction.

  • Adjoint-action invariant: The proof maps a normalized circuit n to the adjoint action adn(Z) and shows this image cannot be proportional to any Pauli matrix.The argument uses parity properties of the coefficients: x0 is odd, while y0 and z0 have opposite parity.
  • Clifford exclusion: Because Clifford conjugation permutes the six vertices ±X, ±Y, and ±Z, excluding Pauli-axis images rules out a normalized circuit evaluating to a Clifford element.The Clifford subgroup acts on su(2) as the octahedral symmetry group.
  • Inductive proof: The parity invariant is proved by induction on T-count, with the base case verified directly and the inductive step preserving the required oddness and opposite-parity conditions.The induction uses the fact that each nontrivial normalized circuit ends in TH or SHTH with a shorter normalized prefix.
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