Matthew Amy’s research while affiliated with Simon Fraser University and other places

The phase folding optimization is a circuit optimization used in many quantum compilers as a fast and effective way of reducing the number of high-cost gates in a quantum circuit. However, existing formulations of the optimization rely on an exact, linear algebraic representation of the circuit, restricting the optimization to being performed on straightline quantum circuits or basic blocks in a larger quantum program. We show that the phase folding optimization can be re-cast as an affine relation analysis , which allows the direct application of classical techniques for affine relations to extend phase folding to quantum programs with arbitrarily complicated classical control flow including nested loops and procedure calls. Through the lens of relational analysis, we show that the optimization can be powered-up by substituting other classical relational domains, particularly ones for non-linear relations which are useful in analyzing circuits involving classical arithmetic. To increase the precision of our analysis and infer non-linear relations from gate sets involving only linear operations — such as Clifford+t — we show that the sum-over-paths technique can be used to extract precise symbolic transition relations for straightline circuits. Our experiments show that our methods are able to generate and use non-trivial loop invariants for quantum program optimization, as well as achieve some optimizations of common circuits which were previously attainable only by hand.

The phase folding optimization is a circuit optimization used in many quantum compilers as a fast and effective way of reducing the number of high-cost gates in a quantum circuit. However, existing formulations of the optimization rely on an exact, linear algebraic representation of the circuit, restricting the optimization to being performed on straightline quantum circuits or basic blocks in a larger quantum program. We show that the phase folding optimization can be re-cast as an \emph{affine relation analysis}, which allows the direct application of classical techniques for affine relations to extend phase folding to quantum \emph{programs} with arbitrarily complicated classical control flow including nested loops and procedure calls. Through the lens of relational analysis, we show that the optimization can be powered-up by substituting other classical relational domains, particularly ones for \emph{non-linear} relations which are useful in analyzing circuits involving classical arithmetic. To increase the precision of our analysis and infer non-linear relations from gate sets involving only linear operations -- such as Clifford+T -- we show that the \emph{sum-over-paths} technique can be used to extract precise symbolic transition relations for straightline circuits. Our experiments show that our methods are able to generate and use non-trivial loop invariants for quantum program optimization, as well as achieve some optimizations of common circuits which were previously attainable only by hand.

Implementations of Roetteler's shifted bent function algorithm have in recent years been used to test and benchmark both classical simulation algorithms and quantum hardware. These circuits have many favorable properties, including a tunable amount of non-Clifford resources and a deterministic output, and moreover do not belong to any class of quantum circuits which is known to be efficiently simulable. We show that this family of circuits can in fact be simulated in polynomial time via symbolic path integrals. We do so by endowing symbolic sums with a confluent rewriting system and show that this rewriting system suffices to reduce the circuit's path integral to the hidden shift in polynomial-time. We hence resolve an open conjecture about the efficient simulability of this class of circuits.

We give a sound and complete equational theory for 3-qubit quantum circuits over the Toffoli-Hadamard gate set $\{ X, CX, CCX, H \}$. That is, we introduce a collection of true equations among Toffoli-Hadamard circuits on three qubits that is sufficient to derive any other true equation between such circuits. To obtain this equational theory, we first consider circuits over the Toffoli-K gate set $\{ X, CX, CCX, K \}$, where $K=H\otimes H$. The Toffoli-Hadamard and Toffoli-K gate sets appear similar, but they are crucially different on exactly three qubits. Indeed, in this case, the former generates an infinite group of operators, while the latter generates the finite group of automorphisms of the well-known $E_8$ lattice. We take advantage of this fact, and of the theory of automorphism groups of lattices, to obtain a sound and complete collection of equations for Toffoli-K circuits. We then extend this equational theory to one for Toffoli-Hadamard circuits by leveraging prior work of Li et al. on Toffoli-Hadamard operators.

The matrices that can be exactly represented by a circuit over the Toffoli-Hadamard gate set are the orthogonal matrices of the form $M/\sqrt{2}{}^k$, where M is an integer matrix and k is a nonnegative integer. The exact synthesis problem for this gate set is the problem of constructing a circuit for a given such matrix. Existing methods produce circuits consisting of $O(2^n\log (n)k)$ gates, where n is the dimension of the matrix. In this paper, we provide two improved synthesis methods. First, we show that a technique introduced by Kliuchnikov in 2013 for Clifford+T circuits can be straightforwardly adapted to Toffoli-Hadamard circuits, reducing the complexity of the synthesized circuit from $O(2^n\log (n)k)$ to $O(n^2\log (n)k)$. Then, we present an alternative synthesis method of similarly improved cost, but whose application is restricted to circuits on no more than three qubits. Our results also apply to orthogonal matrices over the dyadic fractions, which correspond to circuits using the 2-qubit gate $H\otimes H$, rather than the usual single-qubit Hadamard gate H.KeywordsQuantum circuitsExact synthesisToffoli-Hadamard

Vilmart recently gave a complete equational theory for the balanced sum-over-paths over Toffoli-Hadamard circuits, and by extension Clifford+$\mathrm{diag}(1, \zeta_{2^k})$ circuits. Their theory is based on the phase-free ZH-calculus which crucially omits the average rule of the full ZH-calculus, dis-allowing the local summation of amplitudes. Here we study the question of completeness in unbalanced path sums which naturally support local summation. We give a concrete syntax for the unbalanced sum-over-paths and show that, together with symbolic multilinear algebra and the interference rule, various formulations of the average and ortho rules of the ZH-calculus are sufficient to give complete equational theories over arbitrary rings and fields.