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arXiv:quant-ph/0303039 · Quantum Information and Computation

Smaller Circuits for Arbitrary n-qubit Diagonal Computations

Stephen S. Bullock, Igor L. Markov

Published Mar 7, 2003• 13 citations
arXivHTMLPDF

Abstract

A unitary operator U=\sum u_{j,k} |k> ; 0 <= j <= 2^n-1}. These relative phases are often required in applications. Constructing quantum circuits for diagonal computations using standard techniques requires either O(n^2 2^n) controlled-not gates and one-qubit Bloch sphere rotations or else O (n 2^n) such gates and a work qubit. This work provides a recursive, constructive procedure which inputs the matrix coefficients of U and outputs such a diagram containing 2^{n+1}-3 alternating controlled-not gates and one-qubit z-axis Bloch sphere rotations. Up to a factor of two, these circuits are the smallest possible. Moreover, should the computation U be a tensor of diagonal one-qubit computations of the form R_z(α)=e^{-i α/2}|0><0|+ e^{i α/2} |1><1|, then a cancellation of controlled-not gates reduces our circuit to that of an n-qubit tensor.

DOI
10.48550/arxiv.quant-ph/0303039

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A unitary operator U=\sum u_{j,k} |k> ; 0 <= j <= 2^n-1}. These relative phases are often required in applications. Constructing quantum circuits for diagonal computations using standard techniques requires either O(n^2 2^n) controlled-not gates and one-qubit Bloch sphere rotations or else O (n 2^n) such gates and a work qubit. This work provides a recursive, constructive procedure which inputs the matrix coefficients of U and outputs such a diagram containing 2^{n+1}-3 alternating controlled-not gates and one-qubit z-axis Bloch sphere rotations. Up to a factor of two, these circuits are the smallest possible. Moreover, should the computation U be a tensor of diagonal one-qubit computations of the form R_z(α)=e^{-i α/2}|0><0|+ e^{i α/2} |1><1|, then a cancellation of controlled-not gates reduces our circuit to that of an n-qubit tensor.
A unitary operator U = Σj,k uj,k|k〉〈j| is called diagonal when uj,k = 0 unless j = k. The definition extends to quantum computations, where j and k vary over the 2n binary expressions for integers 0,1 ...,2n - 1, given n qubits. Such operators do not affect outcomes of the projective measurement {|j〉〈j|; 0 ≤ j ≤ 2n - 1} but rather create arbitrary relative phases among the computational basis states {|j〉 ; 0 ≤ j ≤ 2n - 1}. Prior quantum circuits for diagonal computations require O(n22n) controlled-not gates and one-qubit Blodl sphere rotations. This work provides a recursive, constructive procedure which inputs U and outputs such a diagram containing 2n+1 - 3 alternating controlled-not gates and one-qubit z- axis Bloch sphere rotations. Up to a factor of two, these circuits are provably the smallest possible. Moreover, the algorithm recognizes certain special case structures within U. For example, if U is a full diagonal tensor of Rz(α) = e-iα/2|0〉〈0| + e-iα/2|1〉〈1| rotations, then a cancellation of controlled-not gates reduces our circuit to the n gate tensor. Moreover, any V ⊕ Rz will output half as many gates as a generic U.
We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.
While the question ``how many CNOT gates are needed to simulate an arbitrary two-qubit operator'' has been conclusively answered -- three are necessary and sufficient -- previous work on this topic assumes that one wants to simulate a given unitary operator up to global phase. However, in many practical cases additional degrees of freedom are allowed. For example, if the computation is to be followed by a given projective measurement, many dissimilar operators achieve the same output distributions on all input states. Alternatively, if it is known that the input state is ∣0⟩\ket{0}∣0⟩, the action of the given operator on all orthogonal states is immaterial. In such cases, we say that the unitary operator is incompletely specified; in this work, we take up the practical challenge of satisfying a given specification with the smallest possible circuit. In particular, we identify cases in which such operators can be implemented using fewer quantum gates than are required for generic completely specified operators.
An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, yyy-rotation, zzz-rotation, and phase gates.
In this article we study n-qubit quantum gates which may be represented as direct sums of two-dimensional unitary operators. We call this class of gates uniformly controlled one-qubit gates. We present a quantum circuit which implements any gate of this type using 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit gates and a single n-qubit diagonal gate. The construction is based on a modified version of the so-called quantum multiplexor circuit. We illustrate the versatility of the uniformly controlled gates by applying them to the decomposition of a general n-qubit gate and an n-qubit state-preparation procedure. Moreover, we study the implementation of these circuits using only nearest-neighbor gates and give the worst-case one-qubit and CNOT gate counts for all of the aforementioned applications. In all four cases, the proposed method either improves or achieves the previously reported upper bounds for the gate counts.