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Universal fault-tolerant quantum computation with only transversal gates and error correction
Adam Paetznick, Ben W. Reichardt
TL;DR
The paper addresses how to achieve universal fault-tolerant quantum computation despite the limits of transversal gates. It uses triorthogonal codes for transversal CCZ and combines this with fault-tolerant H and error correction, while adapting triorthogonal distillation to Toffoli states. The resulting distillation protocol improves the reported T-gate cost for a target error rate, although overhead depends on architectural considerations and triorthogonal-code thresholds.
Problem
Transversal gates alone cannot provide universal quantum computation, motivating constructions that achieve universality without separate resource-state injection and distillation.
Method
The paper uses triorthogonal stabilizer codes with transversal CCZ, a fault-tolerant Hadamard implementation, error correction, and an adapted Toffoli-state distillation procedure.
Results
428.7 average T gates are required for the reported k = 100 hybrid distillation protocol, a 25% saving over the cited procedure alone.
Takeaways & Limitations
A universal fault-tolerant gate set can be built from CCZ and H using triorthogonal codes, and their distillation procedure can reduce Toffoli implementation cost.
Takeaways & Limitations
Triorthogonal-code thresholds are largely unknown and may be lower than those of other codes, while complete overhead depends on architectural considerations.
Abstract
from arXiv · showhide
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controlled-controlled-Z, transversal Hadamard, and fault-tolerant error correction. We also adapt the distillation procedure of Bravyi and Haah to Toffoli gates, improving on existing Toffoli distillation schemes.
STABILIZER CODES BASED ON TRIORTHOGONAL MATRICES
The paper constructs triorthogonal stabilizer codes from binary matrices whose pairwise and triplewise row products have even weight. Even-weight rows define X stabilizers, the orthogonal complement defines Z stabilizers, and odd-weight rows define logical operators.
- Triorthogonal matrices require every distinct pairwise and triplewise row product to have even Hamming weight.
- Even-weight rows of G define X stabilizers, while rows of G⊥ define Z stabilizers.
- Odd-weight rows of G specify the logical X and Z operators of the resulting stabilizer code.
- Fixing the six gauge qubits of the [[15, 7, 3]] Hamming code produces a [[15, 1, 3]] triorthogonal code.
TOFFOLI CONSTRUCTION
The Toffoli construction combines transversal CCZ with a fault-tolerant Hadamard implementation based on stabilizer restoration and X-error correction. Triorthogonality makes transversal CCZ act logically, while the Hadamard procedure restores stabilizers using an encoded |+⟩ ancilla.
- TOFFOLI CONSTRUCTION: The logical Toffoli is obtained from CCZ by conjugating its target qubit with Hadamard gates.
- TOFFOLI CONSTRUCTION: Transversal CCZ implements an encoded CCZ on any triorthogonal code.The proof uses the even-weight triple-product conditions to eliminate unwanted phases, leaving the logical phase only when all three logical inputs are 1.
- TOFFOLI CONSTRUCTION: Transversal H maps logical X to logical Z and vice versa but fails to preserve Z stabilizers associated with G⊥\ G0.
- TOFFOLI CONSTRUCTION: The construction requires a non-self-dual triorthogonal code because no triorthogonal code can support transversal H and CCZ as a universal transversal set.
- TOFFOLI CONSTRUCTION: An encoded |+⟩ ancilla, transversal CNOT, measurement, and X corrections restore the stabilizer group while correcting X errors.
- TOFFOLI CONSTRUCTION: The X-error-correction procedure remains fault tolerant even with additional corrections for Z stabilizers in G⊥\ G0.With k gate failures, the resulting data error has weight at most k for k below half the code distance.
DISCUSSION
The discussion extends transversal fault-tolerant computation to Toffoli-state distillation and higher controlled-Z operations, while identifying overhead and threshold limitations.
- Limitations: The construction may be outperformed near threshold because estimated thresholds for triorthogonal codes are low and resource overhead rises rapidly as physical noise approaches threshold.The paper cites roughly 0.01 percent per gate for the [[15, 1, 3]] code and notes that decomposing CCZ into one- and two-qubit gates may lower the threshold further.
- Toffoli-state distillation: For [[3k + 8, k, 2]] codes, 3k + 8 noisy CCZ gates produce k Toffoli states with error (3k + 1)p^2 to leading order.The input CCZ gates have error rate p, and the Clifford encoding and decoding operations are assumed perfect.
- Toffoli-state distillation: A triorthogonal-code distillation circuit encodes three blocks, applies transversal CCZ gates, and yields k Toffoli states after error detection, decoding, and target Hadamards.The construction uses a triorthogonal code encoding k qubits and accepts outputs conditioned on detecting no errors.
- Overhead comparison: The proposed protocol can reduce Clifford-gate costs, support recursive use, and sit on top of other Toffoli-state distillation protocols, although complete overhead depends on architectural considerations.T-gate cost alone is explicitly identified as an incomplete overhead measure.
- Generalizations: Strengthening the matrix orthogonality conditions makes h-fold controlled-Z gates transversal when all j-tuple products have even weight for 2 ≤ j ≤ h.This generalizes the transversal diagonal-operation construction beyond CCZ.