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arXiv:0706.0479

How to Compile Some NAND Formula Evaluators

Robert R. Tucci

Published Jun 4, 2007• 5 citations
arXivHTMLPDF

Abstract

We say a unitary operator acting on a set of qubits has been compiled if it has been expressed as a SEO (sequence of elementary operations, like CNOTs and single-qubit operations). SEO's are often represented as quantum circuits. arXiv:quant-ph/0702144 by Farhi-Goldstone-Gutmann has inspired a recent flurry of papers, that propose quantum algorithms for evaluating NAND formulas via quantum walks over tree graphs. These algorithms use two types of unitary evolution: oracle and non-oracle. Non-oracle evolutions are independent of the NAND formula input, whereas oracle evolutions depend on this input. In this paper we compile (i.e., give explicit SEOs and their associated quantum circuits for) the oracle and non-oracle evolution operators used in some of these NAND formula evaluators. We consider here only the case of balanced binary NAND trees. Our compilation methods are based on the CSD (Cosine Sine Decomposition), a matrix decomposition from Linear Algebra. The CS decomposition has been used very successfully in the past to compile unstructured unitary matrices exactly.

DOI
10.48550/arxiv.0706.0479

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We say a unitary operator acting on a set of qubits has been compiled if it has been expressed as a SEO (sequence of elementary operations, like CNOTs and single-qubit operations). SEO's are often represented as quantum circuits. arXiv:quant-ph/0702144 by Farhi-Goldstone-Gutmann has inspired a recent flurry of papers, that propose quantum algorithms for evaluating NAND formulas via quantum walks over tree graphs. These algorithms use two types of unitary evolution: oracle and non-oracle. Non-oracle evolutions are independent of the NAND formula input, whereas oracle evolutions depend on this input. In this paper we compile (i.e., give explicit SEOs and their associated quantum circuits for) the oracle and non-oracle evolution operators used in some of these NAND formula evaluators. We consider here only the case of balanced binary NAND trees. Our compilation methods are based on the CSD (Cosine Sine Decomposition), a matrix decomposition from Linear Algebra. The CS decomposition has been used very successfully in the past to compile unstructured unitary matrices exactly.
A quantum compiler is a software program for decomposing ("compiling") an arbitrary unitary matrix into a sequence of elementary operations (SEO). The author of this paper is also the author of a quantum compiler called Qubiter. Qubiter uses a matrix decomposition called the Cosine-Sine Decomposition (CSD) that is well known in the field of Computational Linear Algebra. One way of measuring the efficiency of a quantum compiler is to measure the number of CNOTs it uses to express an unstructured unitary matrix (a unitary matrix with no special symmetries). We will henceforth refer to this number as εεε. In this paper, we show how to improve εεε for Qubiter so that it matches the current world record for εεε, which is held by another quantum compiling algorithm based on CSD.
Quantum Compiling Algorithms decompose (exactly, without approximations) an arbitrary $2^\nb$ unitary matrix acting on $\nb$ qubits, into a sequence of elementary operations (SEO). There are many possible ways of decomposing a unitary matrix into a SEO, and some of these decompositions have shorter length (are more efficient) than others. Finding an optimum (shortest) decomposition is a very hard task, and is not our intention here. A less ambitious, more doable task is to find methods for optimizing small segments of a SEO. Call these methods piecewise optimizations. Piecewise optimizations involve replacing a small quantum circuit by an equivalent one with fewer CNOTs. Two circuits are said to be equivalent if one of them multiplied by some external local operations equals the other. This equivalence relation between circuits has its own class functions, which we call circuit invariants. Dressed CNOTs are a simple yet very useful generalization of standard CNOTs. After discussing circuit invariants and dressed CNOTs, we give some methods for simplifying 2-qubit and 3-qubit circuits. We include with this paper software (written in Octave/Matlab) that checks many of the algorithms proposed in the paper.
A quantum compiling algorithm is an algorithm for decomposing ("compiling") an arbitrary unitary matrix into a sequence of elementary operations (SEO). Suppose UinU_{in}Uin​ is an $\nb$-bit unstructured unitary matrix (a unitary matrix with no special symmetries) that we wish to compile. For $\nb>10$, expressing UinU_{in}Uin​ as a SEO requires more than a million CNOTs. This calls for a method for finding a unitary matrix that: (1)approximates UinU_{in}Uin​ well, and (2) is expressible with fewer CNOTs than UinU_{in}Uin​. The purpose of this paper is to propose one such approximation method. Various quantum compiling algorithms have been proposed in the literature that decompose an arbitrary unitary matrix into a sequence of U(2)-multiplexors, each of which is then decomposed into a SEO. Our strategy for approximating UinU_{in}Uin​ is to approximate these intermediate U(2)-multiplexors. In this paper, we will show how one can approximate a U(2)-multiplexor by another U(2)-multiplexor that is expressible with fewer CNOTs.
Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating nand trees that runs in time O ( √ N log N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N 1/2+ǫ) queries in the conventional quantum query model, for any fixed ǫ> 0. A nand tree of depth n is a balanced binary tree whose internal vertices represent nand gates. Placing bits x1,...,x2 n at the leaves, the root of the nand tree evaluates to the function fn(x1,...,x2 n), where fn: {0, 1} 2n → {0, 1} is defined recursively as follows. For n = 0, f0(x) = x, and for n> 0, fn(x1,..., x2 n) = ¬ ( fn−1(x1,..., x 2 n−1) ∧ fn−1(x 2 n−1 +1,...,x2 n)). (1) The goal of the nand tree problem is to evaluate fn(x1,..., x2 n), making as few queries to the bits x1,..., x2 n as possible. The optimal classical randomized algorithm for this problem makes Θ(N 0.753) queries, where N = 2 n [7, 8, 9]. Until now, no better quantum algorithm was known, whereas the best known quantum lower bound is only Ω ( √ N) [1]. Here we show that for any fixed ǫ> 0, the quantum query complexity of evaluating nand trees is O(N 1/2+ǫ).
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.