Prevailing proposals for the first generation of quantum computers make use of 2-level systems, or qubits, as the fundamental unit of quantum information. However, recent innovations in quantum error correction and magic state distillation protocols demonstrate that there are advantages of using d-level quantum systems, known as \emph{qudits}, over the qubit analogues. When designing a quantum architecture, it is crucial to consider protocols for compilation, the optimal conversion of high-level instructions used by programmers into low-level instructions interpreted by the machine. In this work, we present a general purpose automated compiler for multiqudit exact synthesis based on previous work on qubits that uses an algebraic representation of quantum circuits called phase polynomials. We assume Clifford gates are low-cost and aim to minimise the number of M gates in a Clifford+M circuit, where M is the qudit analog for the qubit T or pi/8 phase gate. A surprising result that showcases our compiler's capabilities is that we found a unitary implementation of the CCZ or Toffoli gate that uses 4 M gates, which compares to 7 T gates for the qubit analogue.
The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by several popular architectures. In addition, I study several questions related to synthesis and optimization of multiqubit operations. \n \n \nWhen studying the single qubit case, I consider two native instruction sets. The first one is Clifford+T; it is supported by conventional quantum computers implementing fault tolerance protocols based on concatenated and surface codes, and by topological quantum computers based on Ising anyons. The second instruction set is the one supported by topological quantum computers based on Fibonacci anyons. I show that in both cases one can use the number theoretic structure of the problem and methods of computational algebraic number theory to achieve improvements over the previous state of the art by factors ranging from 10 to 1000 for instances of the problem interesting in practice. This order of improvement might make certain interesting quantum computations possible several years earlier. \n \n \nThe work related to multiqubit operations is on exact synthesis and optimization of Clifford+T and Clifford circuits. I show an exact synthesis algorithm for unitaries generated by Clifford+T circuits requiring exponentially less number of gates than previous state of the art. For Clifford circuits two directions are studied: the algorithm for finding optimal circuits acting on a small number of qubits and heuristics for larger circuits optimization. The techniques developed allows one to reduce the size of encoding and decoding circuits for quantum error correcting codes by 40-50\\% and also finds their applications in randomized benchmarking protocols.
Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits. First, I introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. I show that these canonical forms are T-optimal and describe an algorithm which takes as input a Clifford+T circuit and outputs the canonical form for that operator. The algorithm runs in time linear in the number of gates of the circuit. 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. Finally, we show that a similar extension of these normal forms to higher dimensions exists, but do not establish uniqueness. Moving to multi-qubit circuits, I provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/√2, i]. We focus on the subrings Z[1/2], Z[1/√2], Z[1/√−2], and 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. I then establish the existence and uniqueness of a normal form for one of these gate sets, the two-qubit gate set of Clifford+Controlled Phase gate CS. This normal form is optimal in the number of CS gates, making it the first normal form that is non-Clifford optimal for a fault tolerant universal multi-qubit gate set. We provide a synthesis algorithm that runs in a time linear in the gate count and outputs the equivalent normal form. In proving the existence and uniqueness of the normal form, we likewise establish the generators and relations for the two-qubit Clifford+CS group. Finally, we demonstrate that a lower bound of 5 log2 (1/ε) + O(1) CS gates are required to ε-approximate any 4 × 4 unitary matrix. Lastly, using the characterization of circuits over the Clifford+CS gate set and the existence of an optimal normal form, I provide an ancilla-free inexact synthesis algorithm for two-qubit unitaries using the Clifford+SC gate set for Pauli-rotations. These operators require 6 log2 (1/ε) + O(1) CS gates to synthesize in the typical case and 8 log2 (1/ε) + O(1) in the worst case.
Before executing a quantum algorithm, one must first decompose the algorithm into machine-level instructions compatible with the architecture of the quantum computer, a process known as quantum compiling. There are many different quantum circuit decompositions for the same algorithm but it is desirable to compile leaner circuits. A fundamentally important cost metric is the
T count -- the number of
T gates in a circuit. For the single qubit case, optimal compiling is essentially a solved problem. However, multi-qubit compiling is a harder problem with optimal algorithms requiring classical runtime exponential in the number of qubits. Here, we present and compare several efficient quantum compilers for multi-qubit Clifford +
T circuits. We implemented our compilers in C++ and benchmarked them on random circuits, from which we determine that our TODD compiler yields the lowest
T counts on average. We also benchmarked TODD on a library of reversible logic circuits that appear in quantum algorithms and found that it reduced the
T count for 97\% of the circuits with an average
T-count saving of 20\% when compared against the best of all previous circuit decompositions.
Abstract Before executing a quantum algorithm, one must first decompose the algorithm into machine-level instructions compatible with the architecture of the quantum computer, a process known as quantum compiling. There are many different quantum circuit decompositions for the same algorithm but it is desirable to compile leaner circuits. A fundamentally important cost metric is the T count—the number of T gates in a circuit. For the single qubit case, optimal compiling is essentially a solved problem. However, multi-qubit compiling is a harder problem with optimal algorithms requiring classical runtime exponential in the number of qubits. Here, we present and compare several efficient quantum compilers for multi-qubit Clifford + T circuits. We implemented our compilers in C++ and benchmarked them on random circuits, from which we determine that our TODD compiler yields the lowest T counts on average. We also benchmarked TODD on a library of reversible logic circuits that appear in quantum algorithms and found that it reduced the T count for 97% of the circuits with an average T -count saving of 20% when compared against the best of all previous circuit decompositions.
Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis
∣0⟩and
∣1⟩states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis
∣0⟩and
∣1⟩states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)
[[9,5,3]]3,
[[5,1,3]]3, and
[[7,1,3]]3codes, and a 9–21% reduction for the ququint (d = 5)
[[10,6,3]]5code when compared to prior work. We also achieved circuit depth reductions up to 42%.
We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of
O(θL2(1−Θ(1/d))pph), while requiring only the ancillary space of a single surface code patch. Here,
θL is the logical rotation angle,
pph is the physical error rate, and
d is the code distance. This scaling marks a significant improvement over the previous state-of-the-art,
O(θLpph), making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for
θL≲10−5, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard
T-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+
T+
φ gate set, rather than the conventional Clifford+
T or Clifford+
φ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of
pph=10−3.