Exact Synthesis of 3-Qubit Quantum Circuits from Non-Binary Quantum Gates Using Multiple-Valued Logic and Group Theory
We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as Controlled-Square-Root-of-Not (i.e. Controlled-V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group th…
Authors: Guowu Yang, William N. N. Hung, Xiaoyu Song
Exact Synthesis of 3-qubit Quantum Circuits from Non-binary Quantum Gates Using Multiple-Valued Logic and Group Theory Guowu Yang, William N. N. Hung, Xiaoyu Song and Marek Perkowski Department of ECE, Portland State University, Portland, Oregon, USA {guowu | whung | song | mperkows} @ ece.pdx.edu Abstract We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as Controlled-Square-Root–of-N ot (i.e. Controlled- V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimization problem to a group permutation problem. The transformation enables us to utilize group theory to exploit the properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we found all reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates. 1. Introduction In this paper, we propose a novel approach to optimally synthesize quantum circuits by group theory where the primary inputs are binary basis states (outputs are not necessarily binary, they may be random binary after measurement of superpositi on states). Finding the smallest number of gates to synthesize a reversible circuit does not necessarily result in a quantum implementation with the lowest cost (in te rms of quantum gates) [2]. The exact minimal costs for NMR [1] realization of several quantum gates from truly quant um (not permutative) gates such as Pauli Rotations or Controlled-Square-Root-of-Not have been calculated [4]. They can be also calculated for other quantum technologies. We focus on synthesizing reversible circuits to quantum implem entations with the lowest cost. The method is general and enumerative. It can be adapted to any particular numerical values of costs. These circuits include common reversible gates that can be used at higher levels of logic synthesis or for technology mapping. We formul ate the quantum logic synthesis problem via group theory. Our m ethod guarantees to find the minimum quantum-cost implementation with truly quantum gates (given a set of specified component gates). In contrast to previous works, which either use permutative reversible gates to design permutative circuits or universal quantum gates to design quantum circuits, we use a subset of quantum gates to design permutative binary circuits that can be either deterministic (when output sy m bols are restricted to basis binary states) or probabilistic (when there is no such constraint imposed on the output symbols). 3. Formulation We briefly describe our problem formulation in this section. Further details of our formulation can be found in our technical report [3]. We are interested in synt hesizing quantum circuits using elementary quantum gates: NOT gates, XOR (controlled-NOT) gates, controlled- V gates and controlled- V + gates. In order to use Group Theory, we need to encode the input values. Given our elementary quantum gates, there are four possible values [2] for each qubit: 0, 1, V 0 , and V 1 . We represent quantum states as permutations (of truth table entries), and quantum gates as permutations as well. The outputs of quantum gates are simply permutations on permutations. For each gate, we construct a banned set as a filter to extract reasonable truth table entries. A banned set N A , for example, is the set of all entries in which the valu e of the controlling qubit A is V 0 or V 1 , because only binary values are allowed to be used for control bits. We constructed a Findi ng Algorithm (see technical report [3] for details) to co mpute the reversible circuit set G[k] of all reversible circuits which have cost k. The idea is to create a set A[k] of all quantum circuits that can be constructed using k or less quantum gates. B[k] is the set of quantum circuits that can be constructed using k (and at least k) quantum gate s. We create pre_G[k]={b’| b’=Restrictedperm(b,S), b % [k]}, where b(S)=S m eans if the input pattern is binary, then its output is also a binary pattern. So the circuit b’ is a reversible circuit with cost k. We create the set G[k] by subtracting G[k-1], …, G[1] from pre_G[k] because when we com pute the Proceedings of the Design, Automation and Test in Europe C onference and Exhibition (DATE’05) 1530-1591/05 $ 20.00 IEEE b’=Restrictedperm(b, S) circuit, b’ may potentially be a mem ber of any G[j], j
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