As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.
We report optimal and asymptotically optimal reversible circuits composed of NOT, CNOT, and Toffoli (NCT) gates, keeping the count by the subsets of the gate types used. This study fine tunes the circuit complexity figures for the realization of reversible functions via reversible NCT circuits. An important consequence is a result on the limitation of the use of the $T$-count quantum circuit metric popular in applications.
The Quantum State Preparation problem aims to prepare an $n$-qubit quantum state $|ψ_v\rangle =\sum_{k=0}^{2^n-1}v_k|k\rangle$ from the initial state $|0\rangle^{\otimes n}$, for a given unit vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\|v\|_2 = 1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth and size complexity remain open when ancillary qubits are available. In this paper, we study efficient constructions of quantum circuits with $m$ ancillary qubits that can prepare…
A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary $n$-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in $O(3^{2\,n} \, \log{1/\varepsilon})$ and ent…
Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；College of Information Engineering, China Jiliang University（中国计量大学信息工程学院）；Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education（教育部计算机网络与信息集成重点实验室（东南大学））；Nanjing Meteorology Bereau（南京市气象局）
The progress in building quantum computers to execute quantum algorithms has recently been remarkable. Grover's search algorithm in a binary quantum system provides considerable speed-up over classical paradigm. Further, Grover's algorithm can be extended to a $d$-ary (qudit) quantum system for utilizing the advantage of larger state space, which helps to reduce the run-time of the algorithm as compared to the traditional binary quantum systems. In a qudit quantum system, an $n$-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this article, a…
Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed. We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons. The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved b…
Learning low dimensional representation is a crucial issue for many machine learning tasks such as pattern recognition and image retrieval. In this article, we present a quantum algorithm and a quantum circuit to efficiently perform A-Optimal Projection for dimensionality reduction. Compared with the best-know classical algorithms, the quantum A-Optimal Projection (QAOP) algorithm shows an exponential speedup in both the original feature space dimension $n$ and the reduced feature space dimension $k$. We show that the space and time complexity of the QAOP circuit are $O\left[ {{{\log }_2}\left…
Variational (or, parameterized) quantum circuits are quantum circuits that contain real-number parameters, that need to be optimized/"trained" in order to achieve the desired quantum-computational effect. For that training, analytic derivatives (as opposed to numerical derivation) are useful. Parameter shift rules have received attention as a way to obtain analytic derivatives, via statistical estimators. In this paper, using Fourier Analysis, we characterize the set of all shift rules that realize a given fixed partial derivative of a multi-parameter VQC. Then, using Convex Optimization, we s…
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) dec…
In circuit QED, a Rabi model can be derived by truncating the Hilbert space of the anharmonic qubit coupled to a linear, reactive environment. This truncation breaks the gauge invariance present in the full Hamiltonian. We analyze the determination of an optimal gauge such that the differences between the truncated and the full Hamiltonian are minimized. Here, we derive a simple criterion for the optimal gauge. We find that it is determined by the ratio of the anharmonicity of the qubit to an averaged environmental frequency. We demonstrate that the usual choices of flux and charge gauge are n…
Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundam…
Gradient descent reliably finds depth- and gate-optimal circuits for generic unitaries
Perimeter Institute for Theoretical Physics（理论物理研究所）；Institute for Quantum Computing, University of Waterloo（量子计算研究所，滑铁卢大学）；Institute for Information Processing (tnt/L3S), Leibniz Universität Hannover, Germany（信息处理研究所（tnt/L3S），莱比锡大学）
Clifford group lies at the core of quantum computation -- it underlies quantum error correction, its elements can be used to perform magic state distillation and they form randomized benchmarking protocols, Clifford group is used to study quantum entanglement, and more. The ability to utilize Clifford group elements in practice relies heavily on the efficiency of their circuit-level implementation. Finding short circuits is a hard problem; despite Clifford group being finite, its size grows quickly with the number of qubits $n$, limiting known optimal implementations to $n{=}4$ qubits. For $n{…
In this note we present explicit canonical forms for all the elements in the two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX, CS]. We provide an algorithm to successively construct the n-qubit CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault-tolerant gate sets.
