arXiv subjects

Wei Zi

Publications and source records attributed to Wei Zi.

11 recordsLinked to original sources

arXiv2026

Quantum Circuit for General Unitary: Improved T-count via Block Flattening and Dilation

Synthesizing arbitrary $n$-qubit unitaries using as few non-Clifford gates as possible is a central problem in fault-tolerant quantum compilation. We present a Clifford+$T$ quantum circuit construction that approximately implements any classically specified unitary to within error $\epsilon$ and achieves a worst-case $T$-count with leading exponential scaling of $2^{5n/4}$ whenever $\log(1/\epsilon)=\operatorname{poly}(n)$. This improves upon the best previous $2^{4n/3}$ scaling. The key innovation lies in treating the target unitary as a single block-encoded object rather than a long product of simpler operations. A technique of block flattening controls the normalization while preserving an efficient implementation of the block encoding; subsequently, quantum singular value transformation maps its common singular value to one, thereby recovering the target unitary.

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arXiv2026

Efficient Depth--Ancilla Tradeoffs for Hamming Weight Computation and Symmetric Boolean Functions

Hamming weight computation maps an $n$-bit input to the number of ones it contains. It is a basic subroutine in quantum computing, and the core building block for symmetric Boolean functions, whose value depends only on the Hamming weight of the input. Moreover, symmetric Boolean functions are among the most common primitives in quantum computing. Efficient circuits for both problems are therefore important for the efficiency of many quantum algorithms. We study the depth-ancilla tradeoffs of Hamming weight computation under two qubit connectivity models, all-to-all and two-dimensional nearest-neighbor square grid (2D), in both the standard and dynamic circuit models. In the standard all-to-all model, we obtain depth $O(\log n)$ with a sublinear number of ancillas. In the standard 2D model, we give a circuit of depth $O(\sqrt n)$ with $O(\log^2 n)$ ancillas, and a matching lower bound showing that $\Theta(\sqrt n)$ is optimal. In both dynamic models, we obtain constant-depth circuits with $O(n^{1+\varepsilon}\operatorname{polylog}\,n)$ ancillary qubits for every fixed $\varepsilon>0$. All constructions give a smooth depth-ancilla tradeoff, and they also extend to arbitrary symmetric Boolean functions.

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arXiv2026

Nearly optimal quantum circuits for Boolean oracles

Quantum oracle of Boolean functions is one of the central bridges between classical and quantum algorithms, but the study focusing at quantum circuit optimization of such oracle is yet closed. In this paper, we propose nearly optimal tradeoffs among circuit size, circuit depth and ancilla count, for quantum oracles of three kinds of Boolean functions: general total Boolean functions with output size $b$: with $1\le m\le\Theta\left(\frac{2^n}{n}\right)$ ancilla, size $\mathcal{O}\left(\frac{b2^n}{\log(n+m)}\right)$, depth $\mathcal{O}\left(\frac{b2^n}{n+m}\right)$; partial Boolean functions of effective support size $d$ and output size $b$: with $\Theta\left(\log d\right)\le m\le \Theta\left(d\right)$ ancilla, size $\mathcal{O}\left(n\log d+bd\right)$, depth $\mathcal{O}\left(\frac{n\log n\log d}{n+m}+\log n+\frac{d(\log d+b\log m)}{m}\right)$; sparse Boolean functions of true input size $d$: with $\Theta\left(\log n+\log d\right)\le m\le\Theta\left(\frac{nd}{\log d}\right)$ ancilla, size $\mathcal{O}\left(n^2\log d+\frac{nd}{\log(\log d+m/n)}\right)$, depth $\mathcal{O}\left(\frac{n^2\log n\log d}{n+m}+\log n+\frac{nd}{m}\right)$. All the size and depth bounds are asymptotically optimal up to logarithmic factors in the corresponding ancilla count regions. We hope these results find applications in scenarios where classical procedures are needed to be embedded into quantum circuits, such as QROM implementation and quantum algorithm design.

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arXiv2026

From Leaves to Clusters: Depth-Efficient SAT-Oracle Synthesis Based on the HRSE Model

Quantum oracles are a common building block of many quantum algorithms, where circuit depth is a primary cost that directly affects overall performance. Synthesizing oracles for SAT (CNF) formulas under a limited ancilla budget, however, tends to yield deep circuits, as existing methods underexploit clause-level parallelism. In this work, we present the Clustered Synthesis Tree (CST), a depth-oriented framework whose core idea is to group the individual clause leaves of a hierarchical synthesis tree into clusters, exposing instance-dependent clause-level parallelism under ancilla constraints. CST comprises three parts: the clause-grouping problem it induces, which we formulate as an ancilla-constrained scheduling problem and prove NP-complete in general, is addressed by SeedGrow, a polynomial-time $O(m^2 k)$ heuristic; ClausePack, a reversible oracle that evaluates a cluster's clauses in parallel at only a logarithmic-depth overhead; and CST-Map, which compiles the clustered tree into an executable SAT-oracle. On random $4$-CNF under the same ancilla budgets, CST reduces the oracle's circuit depth over the state-of-the-art (SOTA) baseline by $68\%$--$94\%$. On the standard SATLIB benchmarks, CST achieves about a $2.6\times$--$43.2\times$ reduction over the SOTA baseline, with the largest gains under dense variable sharing, and matches the baseline's maximum-budget depth using only $3.7\%$--$20\%$ of its ancilla qubits. A Grover-search resource estimate shows the advantage carries over to the full algorithm, reducing total circuit depth by $70\%$--$89\%$.

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arXiv2025

Constant-Depth Quantum Circuits for Arbitrary Quantum State Preparation via Measurement and Feedback

The optimization of quantum circuit depth is crucial for practical quantum computing, as limited coherence times and error-prone operations constrain executable algorithms. Measurement and feedback operations are fundamental in quantum computing (e.g., quantum error correction); we develop a framework using them to achieve constant-depth implementations of essential quantum tasks. This includes preparing arbitrary quantum states with constant-depth circuits through measurement and feedback, breaking the linear-depth lower bound that is required without these operations. Our result paves the way for general quantum circuit compression using measurement and feedback.

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arXiv2024

Shallow Quantum Circuit Implementation of Symmetric Functions with Limited Ancillary Qubits

In quantum computation, optimizing depth and number of ancillary qubits in quantum circuits is crucial due to constraints imposed by current quantum devices. This paper presents an innovative approach to implementing arbitrary symmetric Boolean functions using poly-logarithmic depth quantum circuits with logarithmic number of ancillary qubits. Symmetric functions are those whose outputs rely solely on the Hamming weight of the inputs. These functions find applications across diverse domains, including quantum machine learning, arithmetic circuit synthesis, and quantum algorithm design (e.g., Grover's algorithm). Moreover, by fully leveraging the potential of qutrits (an additional energy level), the ancilla count can be further reduced to 1. The key technique involves a novel poly-logarithmic depth quantum circuit designed to compute Hamming weight without the need for ancillary qubits. The quantum circuit for Hamming weight is of independent interest because of its broad applications, such as quantum memory and quantum machine learning.

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arXiv2024

Efficient Quantum Circuits for Machine Learning Activation Functions including Constant T-depth ReLU

In recent years, Quantum Machine Learning (QML) has increasingly captured the interest of researchers. Among the components in this domain, activation functions hold a fundamental and indispensable role. Our research focuses on the development of activation functions quantum circuits for integration into fault-tolerant quantum computing architectures, with an emphasis on minimizing $T$-depth. Specifically, we present novel implementations of ReLU and leaky ReLU activation functions, achieving constant $T$-depths of 4 and 8, respectively. Leveraging quantum lookup tables, we extend our exploration to other activation functions such as the sigmoid. This approach enables us to customize precision and $T$-depth by adjusting the number of qubits, making our results more adaptable to various application scenarios. This study represents a significant advancement towards enhancing the practicality and application of quantum machine learning.

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arXiv2024

Quantum circuit for multi-qubit Toffoli gate with optimal resource

Resource consumption is an important issue in quantum information processing, particularly during the present NISQ era. In this paper, we investigate resource optimization of implementing multiple controlled operations, which are fundamental building blocks in the field of quantum computing and quantum simulation. We design new quantum circuits for the $n$-Toffoli gate and general multi-controlled unitary, which have only $O(\log n)$-depth and $O(n)$-size, and only require $1$ ancillary qubit. To achieve these results, we explore the potential of ancillary qubits and discover a method to create new conditional clean qubits from existed ancillary qubits. These techniques can also be utilized to construct an efficient quantum circuit for incrementor, leading to an implementation of multi-qubit Toffoli gate with a depth of $O(\log^2n)$ and size of $O(n)$ without any ancillary qubits. Furthermore, we explore the power of ancillary qubits from the perspective of resource theory. We demonstrate that without the assistance of ancillary qubit, any quantum circuit implementation of multi-qubit Toffoli gate must employ exponential precision gates. This finding indicates a significant disparity in computational power of quantum circuits between using and not using ancillary qubits. Additionally, we discuss the comparison of the power of ancillary qubits and extra energy levels in quantum circuit design.

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arXiv2023

Optimal Synthesis of Multi-Controlled Qudit Gates

We propose a linear-size synthesis of the multi-controlled Toffoli gate on qudits with at most one borrowed ancilla. This one ancilla can even be saved when the qudit dimension is odd. Our synthesis leads to improvements in various quantum algorithms implemented on qudits. In particular, we obtain (i) a linear-size and one-clean-ancilla synthesis of multi-controlled qudit gates; (ii) an optimal-size and one-clean-ancilla synthesis of unitaries on qudits; (iii) a near-optimal-size and ancilla-free/one-borrowed-ancilla implementation of classical reversible functions as qudit gates.

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arXiv2022

Efficient quantum circuit synthesis for SAT-oracle with limited ancillary qubit

How to implement quantum oracle with limited resources raises concerns these days. We design two ancilla-adjustable and efficient algorithms to synthesize SAT-oracle, the key component in solving SAT problems. The previous work takes 2m-1 ancillary qubits and O(m) elementary gates to synthesize an m clauses oracle. The first algorithm reduces the number of ancillary qubits to 2\sqrt{m}, with at most an eightfold increase in circuit size. The number of ancillary qubits can be further reduced to 3 with a quadratic increase in circuit size. The second algorithm aims to reduce the circuit depth. By leveraging of the second algorithm, the circuit depth can be reduced to O(log m) with m ancillary qubits.

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arXiv2019

Cake Cutting on Graphs: A Discrete and Bounded Proportional Protocol

The classical cake cutting problem studies how to find fair allocations of a heterogeneous and divisible resource among multiple agents. Two of the most commonly studied fairness concepts in cake cutting are proportionality and envy-freeness. It is well known that a proportional allocation among $n$ agents can be found efficiently via simple protocols [16]. For envy-freeness, in a recent breakthrough, Aziz and Mackenzie [5] proposed a discrete and bounded envy-free protocol for any number of players. However, the protocol suffers from high multiple-exponential query complexity and it remains open to find simpler and more efficient envy-free protocols. In this paper we consider a variation of the cake cutting problem by assuming an underlying graph over the agents whose edges describe their acquaintance relationships, and agents evaluate their shares relatively to those of their neighbors. An allocation is called locally proportional if each agent thinks she receives at least the average value over her neighbors. Local proportionality generalizes proportionality and is in an interesting middle ground between proportionality and envy-freeness: its existence is guaranteed by that of an envy-free allocation, but no simple protocol is known to produce such a locally proportional allocation for general graphs. Previous works showed locally proportional protocols for special classes of graphs, and it is listed in both [1] and [8] as an open question to design simple locally proportional protocols for more general classes of graphs. In this paper we completely resolved this open question by presenting a discrete and bounded locally proportional protocol for any given graphs. Our protocol has a query complexity of only single exponential, which is significantly smaller than the six towers of $n$ query complexity of the envy-free protocol given in [5].

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