Quantum computers promise exponential speedups for problems in cryptography, chemistry, and optimization. Realizing this promise requires fault tolerance: physical qubits are noisy, so logical qubits must be encoded redundantly across many physical ones using quantum error-correcting codes. In most practical fault-tolerance schemes, T gates cannot be implemented transversally and instead require costly magic-state distillation protocols involving a complex set of operations. As a result, T-gate count can dominate the resource budget of large-scale quantum computations, making T-count minimization a central bottleneck on the path to quantum advantage. Existing T-count optimization tools, however, do not scale to the circuits that quantum advantage demands.
We present theoretical and practical results on T-gate optimization. On the theoretical side, we give a linear-time randomized algorithm for phase folding, based on a novel randomized static analysis. Our static analysis soundly approximates the set of reachable quantum states with an arbitrarily high probability. Our key insight is a static analysis that does not track symbolic expressions, but propagates constant-width bitstrings down the circuit. On the practical side, our implementation, TZAP, is multiple orders of magnitude faster than state-of-the-art tools -- such as PyZX, VOQC, and Feynman -- closely matches their T-count reductions on standard benchmarks, and within seconds on a laptop computer can optimize circuits with millions of gates.
Quantum computers promise exponential speedups for problems in cryptography, chemistry, and optimization. Realizing this promise requires fault tolerance: physical qubits are noisy, so logical qubits must be encoded redundantly across many physical ones using quantum error-correcting codes. In most practical fault-tolerance schemes, T gates cannot be implemented transversally and instead require costly magic-state distillation protocols involving a complex set of operations. As a result, T-gate count can dominate the resource budget of large-scale quantum computations, making T-count minimization a central bottleneck on the path to quantum advantage. Existing T-count optimization tools, however, do not scale to the circuits that quantum advantage demands.
We present theoretical and practical results on T-gate optimization. On the theoretical side, we give a linear-time randomized algorithm for phase folding, based on a novel randomized static analysis. Our static analysis soundly approximates the set of reachable quantum states with an arbitrarily high probability. Our key insight is a static analysis that does not track symbolic expressions, but propagates constant-width bitstrings down the circuit. On the practical side, our implementation, TZAP, is multiple orders of magnitude faster than state-of-the-art tools -- such as PyZX, VOQC, and Feynman -- closely matches their T-count reductions on standard benchmarks, and within seconds on a laptop computer can optimize circuits with millions of gates.
Quantum Error Correction (QEC) is the corner-stone of practical Fault-Tolerant Quantum Computing (FTQC), but incurs enormous resource overheads. Circuits must decompose into Clifford +T gates, and the non-transversal T gates demand costly magic-state distillation. As circuit complexity grows, sequential T-gate layers (” T-depth”) increase, amplifying the spatiotemporal overhead of QEC. Optimizing T-depth is NP-hard, and existing greedy or brute-force strategies are either inefficient or computationally prohibitive. We frame T-depth reduction as a search optimization problem and present a Genetic Algorithm (GA) framework that approximates optimal layer-merge patterns across the non-convex search space. We introduce a mathematical formulation of the circuit expansion for systematic layer reordering and a greedy initial merge-pair selection, accelerating the convergence and enhancing the solution quality. In our benchmark with ∼90−100 qubits, our method reduces T-depth by 79.23% and overall T-count by 41.86%. Compared to the standard reversible circuit benchmarks, we achieve a ∼2.58× average improvement in T-depth over the state-of-the-art methods, demonstrating its viability for near-term FTQC.
Quantum computation promises exponential or polynomial speedups over classical computation for a wide range of problems, including integer factoring, quantum chemistry, and linear-system solving.However, current quantum devices remain too noisy to execute complex algorithms reliably.Overcoming this noise and achieving quantum advantage require a fault-tolerant quantum computer (FTQC).The resource overhead associated with fault-tolerant operation is substantial, and reducing this overhead remains one of the central challenges.A major source of this overhead arises from the implementation of non-Clifford gates, which are essential for universal quantum computation.A non-Clifford gate such as the T gate is implemented using a special quantum state known as a magic state, which is prepared via a process called magic state distillation.Magic state distillation is resource-intensive and remains a major bottleneck in FTQC.Hence, reducing the distillation overhead is necessary to lower the overall FTQC overhead.The large overhead of magic state distillation has significantly influenced FTQC architectures.In particular, it has motivated compilation schemes optimized under the constraint of expensive magic state distillation.While reducing the distillation overhead is crucial, it is not sufficient; lowering the overall FTQC overhead also requires revisiting design choices guided by this constraint.Therefore, efficient methods are required not only for preparing magic states but also for consuming them.In this thesis, we reduce the overhead of fault-tolerant quantum computation by focusing on the preparation and consumption of magic states.In contrast to traditional logical-level distillation, physical-level distillation allows direct implementation with physical gates and has the potential to drastically reduce the distillation overhead.However, existing physical-level distillation protocols, such as zero-level distillation, cannot directly yield high-fidelity magic states.We propose two magic state distillation protocols that use physical-level distillation techniques to efficiently prepare high-fidelity magic states.The first protocol is a two-level distillation protocol using physical-level distillation as the first-level stage.The second protocol utilizes lattice surgery to reduce the spacetime overhead of a physical-level distillation protocol.The former targets a high-fidelity regime, whereas the latter targets an intermediate-fidelity regime-lower than the former, but higher than zero-level distillation.Next, we investigate FTQC architectures that can efficiently consume the magic states prepared by these protocols.Because magic state distillation is probabilistic, occasional failures can delay computation.Moreover, physical-level distillation protocols generally exhibit higher failure probabilities than logical-level ones.This delay is amplified by parallel execution, making it an i ii important consideration for FTQC using physical-level distillation protocols.We therefore propose a scheme to mitigate the negative performance impact caused by magic state distillation failures by introducing a pool of magic states for each distillation factory.The proposed distillation protocols allow the simultaneous preparation of many magic states.Leveraging this capability, we propose an FTQC compilation scheme optimized for highly parallel quantum algorithms.By performing magic state distillation locally and consuming magic states in parallel, the scheme enables highly parallel execution and consequently reduces the computational time overhead.Overall, this thesis presents a unified approach to reducing the overhead of fault-tolerant quantum computation.By improving both the preparation and consumption of magic states, it advances the design of resource-efficient FTQC architectures.These results demonstrate that efficient magic state preparation motivates efficient magic state consumption, and vice versa.Together, they bring us closer to the practical realization of fault-tolerant quantum computers.
Fault-tolerant quantum computing typically requires the transpilation of arbitrary quantum circuits into a finite, universal gate set, such as Clifford+T. As a baseline, Diagonal approximation can be used for synthesizing single-qubit Pauli rotations, yielding an approximating sequence with T-count that equals 3log2(1/ε) for a target precision ε. Magnitude Approximation can reduce the T-count to only 1log2(1/ε) by allowing large residual errors, which are rotations about orthogonal axes. Within a complete quantum circuit, these residual errors can then be absorbed into neighboring gates before they are approximated themselves. Determining the optimal allocation of approximation strategies within a large, multi-qubit circuit presents a significant combinatorial challenge. In this work, we present a linear-time algorithm that guarantees an optimal solution to this problem. We demonstrate that the issue of delegating Magnitude versus Diagonal approximation across a circuit maps formally to a classical 1D Ising model with a spatially varying field. By minimizing the energy of this Hamiltonian, we identify the optimal approximation configuration for each rotation without exponential overhead. Benchmarking our method against standard diagonal approximation on random quantum circuits, we observe an average reduction of 26\% in the total approximating circuit gate count, offering a significant efficiency gain for the implementation of quantum algorithms on near-term and fault-tolerant architectures.
Fault-tolerant architectures implement non-Clifford T gates through magic-state distillation, so the T-count of a synthesized circuit dominates its physical cost. The Solovay-Kitaev algorithm approximates any single-qubit unitary from a finite gate set with a sequence length that grows only polylogarithmically in the inverse target error, but it optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see. We report a measurement of what diagrammatic post-processing recovers from that redundancy. Twelve hundred random single-qubit targets, spanning the three Pauli rotation families and the general gate U(theta, phi, lambda), are synthesized over Clifford+T at three recursion depths, translated into graph-like ZX-diagrams, simplified by automated rewriting, and extracted back to circuits. Post-processing removes 26.6-30.1% of the total gate count and 18.5-22.2% of the T-count. The absolute saving grows with recursion depth, from about 60 to about 1600 gates, while the fractional saving does not: it rises slightly from the shallowest setting and is then flat across a twenty-five-fold change in circuit length, and by the deepest setting the four target families are no longer distinguishable from one another. Because the rewrite rules preserve the implemented linear map, the approximation error is unchanged. The compile-time cost of the rewriting layer, by contrast, grows sharply with depth and comes to dominate the synthesis itself.
Simulating quantum error correction (QEC) circuits including non-Clifford gates at scale is important to accelerate progress toward fault-tolerant quantum computing. Here we demonstrate that matrix product state (MPS) techniques can handle many QEC circuits exactly and without restriction on gate types. Crucially, we find that MPS efficiency depends sensitively on implementation choices, and we introduce a series of targeted optimizations that reduce bond dimensions and simulation time by several orders of magnitude compared to naive approaches. We illustrate this with examples including: (a) a rotated surface code quantum memory up to distance 11, (b) logical Bell-state preparation up to distance 9, (c) a 15-to-1 magic-state distillation circuit including hundreds of QEC rounds that we optimize to be simulated with only 11 logical qubits (187 physical qubits) and a maximal bond dimension of 64 in under 40 seconds, and (d) a narrow, deep random circuit that scales linearly with the number of T gates. These results demonstrate the importance of circuit-level optimizations and position MPS as a valuable complement to near-Clifford simulators for QEC circuits.
The quantum Fourier transform (QFT) is a fundamental component in various quantum algorithms, including Shor's factoring algorithm and the Harrow-Hassidim-Lloyd (HHL) algorithm for solving systems of linear equations. Efficient implementation of the QFT is essential for the practical realization of large-scale quantum algorithms, especially in fault-tolerant quantum computing. In fault-tolerant implementations, the Clifford + T gate library is the standard choice for building quantum circuits. As the most resource-intensive component within this framework, the T gate's associated cost poses a significant challenge to the efficient implementation of the QFT and its dependent algorithms. While approximate QFT (AQFT) circuits reduce this cost, state-of-the-art implementations still require a T-count of [Formula: see text] and a T-depth of [Formula: see text]. Although these results represent a notable achievement, the associated resource cost remains a primary bottleneck for practical, large-scale quantum algorithms, motivating further optimization. To address this bottleneck, this paper introduces two novel [Formula: see text]-qubit AQFT circuits with an approximation error of [Formula: see text]. Our first design, AQFT Circuit 1, halves the T-count to [Formula: see text] by constructing inverse phase gradient transformation (PGT) circuits without using additional non-Clifford gates and by implementing the inverse PGTs using quantum adders. Our second design, AQFT Circuit 2, reduces the T-depth to [Formula: see text] through parallelization of the inverse PGTs that add only [Formula: see text] additional T gates. For both AQFT circuits, the state-of-the-art linear-depth quantum adder is employed. We demonstrate that employing the linear-depth quantum adder provides advantages over the currently known logarithmic-depth quantum adder, not only in terms of T-count but also in T-depth optimization for the AQFT, particularly within the range [Formula: see text], which encompasses practical system sizes.