arXiv subjects

Andrew N. Glaudell

Publications and source records attributed to Andrew N. Glaudell.

9 recordsLinked to original sources

arXiv2026

High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

Exact synthesis provides unconditional optimality and canonical structure, but is often limited to small, carefully scoped regimes. We present an exact synthesis framework for two-qubit circuits over the Clifford+$T$ gate set that optimizes $T$-count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.

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arXiv2025

Computing Efficiently in QLDPC Codes

It is the prevailing belief that quantum error correcting techniques will be required to build a utility-scale quantum computer able to perform computations that are out of reach of classical computers. The QECCs that have been most extensively studied and therefore highly optimized, surface codes, are extremely resource intensive in terms of the number of physical qubits needed. A promising alternative, QLDPC codes, has been proposed more recently. These codes are much less resource intensive, requiring significantly fewer physical qubits per logical qubit than practical surface code implementations. A successful application of QLDPC codes would therefore drastically reduce the timeline to reaching quantum computers that can run algorithms with proven exponential speedups like Shor's algorithm and QPE. However to date QLDPC codes have been predominantly studied in the context of quantum memories; there has been no known method for implementing arbitrary logical Clifford operators in a QLDPC code proven efficient in terms of circuit depth. In combination with known methods for implementing T gates, an efficient implementation of the Clifford group unlocks resource-efficient universal quantum computation. In this paper, we introduce a new family of QLDPC codes that enable efficient compilation of the full Clifford group via transversal operations. Our construction executes any m-qubit Clifford operation in at most O(m) syndrome extraction rounds, significantly surpassing state-of-the-art lattice surgery methods. We run circuit-level simulations of depth-126 logical circuits to show that logical operations in our QLDPC codes attains near-memory performance. These results demonstrate that QLDPC codes are a viable means to reduce the resources required to implement all logical quantum algorithms, thereby unlocking a reduced timeline to commercially valuable quantum computing.

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arXiv2024

Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

It is known that the matrices that can be exactly represented by a multiqubit circuit over the Toffoli+Hadamard, Clifford+$T$, or, more generally, Clifford-cyclotomic gate set are precisely the unitary matrices with entries in the ring $\mathbb{Z}[1/2,ζ_k]$, where $k$ is a positive integer that depends on the gate set and $ζ_k$ is a primitive $2^k$-th root of unity. In the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree $3^k$ by extending the classical qutrit gates $X$, $CX$, and $CCX$ with the Hadamard gate $H$ and the $T_k$ gate $T_k=\mathrm{diag}(1,ω_k, ω_k^2)$, where $ω_k$ is a primitive $3^k$-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when $k=1$, and to the qutrit Clifford+$T_k$ gate set when $k>1$. We then prove that a $3^n\times 3^n$ unitary matrix $U$ can be represented by an $n$-qutrit circuit over the Clifford-cyclotomic gate set of degree $3^k$ if and only if the entries of $U$ lie in the ring $\mathbb{Z}[1/3,ω_k]$.

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arXiv2024

Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits

Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,ζ_n)$, where $ζ_n$ is a primitive $n$-th root of unity. In this note, we show that, when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $\mathcal{G}_n$ if and only if the entries of $U$ belong to the ring $\mathbb{Z}[1/2,ζ_n]$. We moreover show that $\log(n)-2$ ancillas are always sufficient to construct a circuit for $U$. Our results generalize prior work to an infinite family of gate sets and show that the limitations that apply to single-qubit unitaries, for which the correspondence between Clifford-cyclotomic operators and matrices over $\mathbb{Z}[1/2,ζ_n]$ fails for all but finitely many values of $n$, can be overcome through the use of ancillas.

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arXiv2023

Improved Synthesis of Toffoli-Hadamard Circuits

The matrices that can be exactly represented by a circuit over the Toffoli-Hadamard gate set are the orthogonal matrices of the form $M/ \sqrt{2}{}^k$, where $M$ is an integer matrix and $k$ is a nonnegative integer. The exact synthesis problem for this gate set is the problem of constructing a circuit for a given such matrix. Existing methods produce circuits consisting of $O(2^n \log(n)k)$ gates, where $n$ is the dimension of the matrix. In this paper, we provide two improved synthesis methods. First, we show that a technique introduced by Kliuchnikov in 2013 for Clifford+$T$ circuits can be straightforwardly adapted to Toffoli-Hadamard circuits, reducing the complexity of the synthesized circuit from $O(2^n \log(n)k)$ to $O(n^2 \log(n)k)$. Then, we present an alternative synthesis method of similarly improved cost, but whose application is restricted to circuits on no more than three qubits. Our results also apply to orthogonal matrices over the dyadic fractions, which correspond to circuits using the 2-qubit gate $H\otimes H$, rather than the usual single-qubit Hadamard gate $H$.

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arXiv2021

Optimal Two-Qubit Circuits for Universal Fault-Tolerant Quantum Computation

We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of 5log(1/epsilon)+O(1) on the number of CS gates required to epsilon-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.

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arXiv2020

Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+$T$ circuits by considering unitary matrices over subrings of $\mathbb{Z}[1/\sqrt{2},i]$. We focus on the subrings $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$, and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates $\{X, CX, CCX\}$ with an analogue of the Hadamard gate and an optional phase gate.

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arXiv2019

Canonical forms for single-qutrit Clifford+T operators

We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits.

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arXiv2015

Serialized Quantum Error Correction Protocol for High-Bandwidth Quantum Repeaters

Advances in single photon creation, transmission, and detection suggest that sending quantum information over optical fibers may have losses low enough to be correctable using a quantum error correcting code. Such error-corrected communication is equivalent to a novel quantum repeater scheme, but crucial questions regarding implementation and system requirements remain open. Here we show that long range entangled bit generation with rates approaching $10^8$ ebits/s may be possible using a completely serialized protocol, in which photons are generated, entangled, and error corrected via sequential, one-way interactions with a minimal number of matter qubits. Provided loss and error rates of the required elements are below the threshold for quantum error correction, this scheme demonstrates improved performance over transmission of single photons. We find improvement in ebit rates at large distances using this serial protocol and various quantum error correcting codes.

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