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A complete transformation rule set and a minimal equation set for CNOT-based 3 qubits quantum circuits

Issei Sakashita

Published Apr 1, 2013• 0 citations
OpenAlex

Abstract

We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to a same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002 [IKY02], we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rule set consists of 114 rules. Moreover, we found a minimal subset of equations for the initial equation set.

Related papers

A complete transformation rule set and a minimal equation set for CNOT-based 3-qubit quantum circuits (Draft)

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This paper gives a simple but nontrivial set of local transformation rules for Control-NOT(CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S1 and S2, there is a sequence of transformations, each of them in the rule set, which changes S1 to S2. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT based circuits. As a preliminary example, we give a design procedure based on our transformation rules which reduces the cost of CNOT-based circuits.

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This paper gives a simple but nontrivial set of local transformation rules for control-NOT (CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S/sub 1/ and S/sub 2/, there is a sequence of transformations, each of them in the rule set, which changes S/sub 1/ to S/sub 2/. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT-based circuits. As a preliminary example, we give a design procedure based on the transformation rules which reduces the cost of CNOT-based circuits.

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We introduce the first minimal and complete equational theory for quantum circuits. Hence, we show that any true equation on quantum circuits can be derived from simple rules, all of them being standard except a novel but intuitive one which states that a multi-control 2π rotation is nothing but the identity. Our work improves on the recent complete equational theories for quantum circuits, by getting rid of several rules including a fairly impractical one. One of our main contributions is to prove the minimality of the equational theory, i.e. none of the rules can be derived from the other ones. More generally, we demonstrate that any complete equational theory on quantum circuits (when all gates are unitary) requires rules acting on an unbounded number of qubits. Finally, we also simplify the complete equational theories for quantum circuits with ancillary qubits and/or qubit discarding.

arXiv:2311.07476•ACM/IEEE Symposium on Logic in Computer Science (LICS)•Jun 21, 2024•3 citations
We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to a same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002 [IKY02], we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of termination' and confluence'). Our transformation rule set consists of 114 rules. Moreover, we found a minimal subset of equations for the initial equation set.
We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to the same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002, we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rule set consists of 114 rules. Moreover, we discovered a minimal combination of equations for the initial equation set.
This paper gives a simple but nontrivial set of local transformation rules for Control-NOT(CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S1 and S2, there is a sequence of transformations, each of them in the rule set, which changes S1 to S2. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT based circuits. As a preliminary example, we give a design procedure based on our transformation rules which reduces the cost of CNOT-based circuits.
Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. A finite rule set is most informative when it separates the genuine algebra of a circuit fragment from the structural treatment of wires. This paper gives six near-Clifford fragments a common PROP treatment, where wire permutations are structural: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, and qutrit Clifford. Starting from prior completeness theorems, we transfer completeness into this setting and remove redundant non-structural rules, then check minimality by separating interpretations tailored to individual axioms; the resulting presentations are minimal in all arities for qubit Clifford, real Clifford, and CNOT-dihedral, minimal in bounded ranges for the remaining fragments, and comparable by one transfer-and-separation pattern.
This paper gives a simple but nontrivial set of local transformation rules for control-NOT (CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S/sub 1/ and S/sub 2/, there is a sequence of transformations, each of them in the rule set, which changes S/sub 1/ to S/sub 2/. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT-based circuits. As a preliminary example, we give a design procedure based on the transformation rules which reduces the cost of CNOT-based circuits.
Reversible logic synthesis is a crucial component in quantum electronic design automation. While rule-based methodologies have gained prominence in reversible circuit optimization, the completeness of the transformation rule systems is a longstanding problem in this domain. In this work, we propose the first complete set of transformation rules for reversible circuits, comprising five fundamental rules: any two equivalent reversible circuits can be transformed into each other using the rules. To prove the completeness, a canonical circuit representation for reversible functions is introduced, and we show that every reversible function is computed by a unique reversible circuit in the canonical form, and any reversible circuit can be transformed into its canonical form by applying the rules.
We introduce the first minimal and complete equational theory for quantum circuits. Hence, we show that any true equation on quantum circuits can be derived from simple rules, all of them being standard except a novel but intuitive one which states that a multi-control 2π rotation is nothing but the identity. Our work improves on the recent complete equational theories for quantum circuits, by getting rid of several rules including a fairly impractical one. One of our main contributions is to prove the minimality of the equational theory, i.e. none of the rules can be derived from the other ones. More generally, we demonstrate that any complete equational theory on quantum circuits (when all gates are unitary) requires rules acting on an unbounded number of qubits. Finally, we also simplify the complete equational theories for quantum circuits with ancillary qubits and/or qubit discarding.