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arXiv:1511.05758 · Physical Review A

Phase-context decomposition of diagonal unitaries for higher-dimensional systems

Kerstin Beer, Friederike Anna Dziemba

Published May 25, 2016• 3 citations
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Abstract

We generalize the efficient decomposition method for phase-sparse diagonal operators of J. Welch et al. [Quantum Info. Comput. 16, 87 (2016)] to qudit systems. The phase-context-aware method focuses on cascaded entanglers, whose decomposition into multicontrolled inc gates can be optimized by the choice of a proper signed base-$d$ representation for the natural numbers. While the gate count of the best-known decomposition method for general diagonal operators on qubit systems scales with $O({2}^{n})$, the circuits synthesized by the Welch algorithm for diagonal operators with $k$ distinct phases are upper-bounded by $O({n}^{2}k)$, which is generalized to $O(d{n}^{2}k)$ for the qudit case in this paper.

DOI
10.1103/physreva.93.052333

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We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+TTT basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit. The equivalent TTT-count of the synthesized circuit is bounded by k C0log⁡2(1/ε)+E(n,k)k \, C_0 \log_2(1/\varepsilon) + E(n,k)kC0​log2​(1/ε)+E(n,k), where kkk is the number of distinct phases in the diagonal nnn-qubit unitary, ε\varepsilonε is the desired precision, C0C_0C0​ is a quality factor of the implementation method ($1
Multicontrolled unitary gates have been a subject of interest in quantum computing since their conception and are widely used in quantum algorithms. The current state-of-the-art approach to implementingnnn-qubit multicontrolled gates with a single target without relying on auxiliary qubits or approximate results involves the use of a quadratic number of single-qubit and CNOT gates. However, linear solutions are possible for the case where the controlled gate is special unitary, SU(2). The decomposition of annnn-qubit multicontrolled SU(2) gate requires a circuit with a number of CNOT gates proportional to28n28n28n. In this work, we present a new decomposition ofnnn-qubit multicontrolled SU(2) gates that require a circuit with a number of CNOT gates proportional to20n20n20nand proportional to16n16n16nif the SU(2) gate has at least one real-valued diagonal. The proposed algorithms produce the most efficient known circuits and improve the existing algorithm by reducing the number of CNOT gates and the overall circuit depth. As an application, we show the use of this decomposition for sparse quantum state preparation. Our results are further validated by demonstrating a proof of principle on a quantum device accessed through quantum cloud services.
We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+T basis, that minimizes the number of phase-rotation gates in the synthesized approximation circuit. The equivalent T-count of the synthesized circuit is bounded by kC_0 log_2 (1/ε)+E(n, k), where k is the number of distinct phases in the diagonal n-qubit unitary, ε is the desired precision, C0 is a quality factor of the implementation method (1 < C_0 < 4), and E(n, k) is the total entanglement cost (in T gates). We determine an optimal decision boundary in (n, k, ε)-space where our decomposition algorithm achieves lower entanglement cost than previous state-of-the-art techniques. Our method outperforms state-of-the-art techniques for a practical range of ε values and diagonal operators and can reduce the number of T gates exponentially in n when k far less than 2^ n .
Current quantum devices have unutilized high-level quantum resources. More and more attention has been paid to the qudit quantum systems with larger than two dimensions to maximize the potential computing power of quantum computation. Then, a natural problem arises: How do we implement quantum algorithms on qudit quantum systems? In this work, we propose a novel qudit phase gadget method for synthesizing the qudit diagonal unitary matrices. This method is suitable for the Noisy Intermediate-Scale Quantum (NISQ) and fault-tolerant eras due to its versatility in different connectivity architectures and the optimality of its resource consumption. The method can work on any connectivity architecture with asymptotic optimal circuit depth and size. For a 10-qutrit diagonal unitary, our algorithm reduces the circuit depth form about 100000 to 500 with 300 ancillary qutrits. Further, this method can be promoted to different quantum circuit synthesis problems, such as quantum state preparation problems, general unitary synthesis problems, etc.
Circuit synthesis of multi-controlled gates is crucial for qudit (ddd-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general (n−1)(n-1)(n−1)-controlled unitaries on nnn qudits, we reduce the controlled-increment (CINC) and generalized controlled-XXX (GCX) gate counts to O(n2)O(n^2)O(n2), improving upon existing O(n2+log⁡2d)O(n^{2+\log_2 d})O(n2+log2​d) CINC and O(n3)O(n^3)O(n3) GCX bounds. For (n−1)(n-1)(n−1)-controlled special unitaries, this complexity is further reduced to O(n)O(n)O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from nnn to mmm qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when ddd is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general nnn-qudit unitaries.
Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by unitary matrices, and therefore for a quantum system consists of multiple qudits, a gate may be synthesized by matrix decomposition techniques such as QR factorization and the Cosine-sine Decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid and d-valued quantum circuits.