Robust quantum compilation and circuit optimisation via energy minimisation

Tyson Jones and Simon C. Benjamin

Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

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We explore a method for automatically recompiling a quantum circuit $\mathcal{A}$ into a target circuit $\mathcal{B}$, with the goal that both circuits have the same action on a specific input i.e. $\mathcal{B}{\mid{in}\rangle}=\mathcal{A}{\mid{in}\rangle}$. This is of particular relevance to hybrid, NISQ-era algorithms for dynamical simulation or eigensolving. The user initially specifies $\mathcal{B}$ as a blank template: a layout of parameterised unitary gates configured to the identity. The compilation then proceeds using quantum hardware to perform an isomorphic energy-minimisation task, and an optional gate elimination phase to compress the circuit. If $\mathcal{B}$ is insufficient for perfect recompilation then the method will result in an approximate solution. We optimise using imaginary time evolution, and a recent extension of quantum natural gradient for noisy settings. We successfully recompile a $7$-qubit circuit involving $186$ gates of multiple types into an alternative form with a different topology, far fewer two-qubit gates, and a smaller family of gate types. Moreover we verify that the process is $robust$, finding that per-gate noise of up to $1\%$ can still yield near-perfect recompilation. We test the scaling of our algorithm on up to $20$ qubits, recompiling into circuits with up to $400$ parameterized gates, and incorporate a custom adaptive timestep technique. We note that a classical simulation of the process can be useful to optimise circuits for today's prototypes, and more generally the method may enable `blind' compilation i.e. harnessing a device whose response to control parameters is deterministic but unknown.

The code and resources used to generate our results are openly available online [1] [2]. A simple Mathematica demonstration of our algorithm can be found at

To run a program, a computer needs to understand how to perform each operation written in the code. But there are possibly thousands of programming languages which contain extremely many basic operations. How can one computer run them all? It can do so using compilers: Software written in a language the computer understands which can translate programs into instructions the computer knows how to carry out. Compilers can often also optimise the program, by reducing the number of basic operations yet still obtain the same final result. You’re probably viewing this webpage with a device running tens of compilers at once!

But quantum computers run a different kind of program. Quantum programs are written and understood in a very different way to the programs running on your home computer. They therefore need very different methods to compile them, that is shorten them or re-express them using a different set of operations. Many existing quantum compilers use clever maths to find alternate sequences of operations which have an identical effect on the quantum state. However, they run on classical computers, and require the result of these operations on the quantum state are precisely known and are storable in the computer.

But on noisy, near-future quantum computers, we often don’t know precisely what our elementary operations are doing to the quantum state. Sometimes we wish re-express our programs using a new set of operations which cannot precisely recreate the effect on the quantum state, but can bring us pretty close. How can we approximately translate our program?

In this work, we introduce a new method for performing approximate compilation to optimise the programs written for near-future quantum computers. It recasts the problem of compiling into one of finding the lowest energy state of a quantum system, for which recent developments have shown that near-future quantum computers can be quite good at. Our algorithm can run on real quantum hardware, or be simulated on a classical machine, in order to shorten quantum programs or translate them into different families of instructions. We explore one fascinating application of our recompilation method – to prolong the accuracy of physical simulation circuits by recompiling them on the fly! In line with our open science philosophy, we share all our code to simulate our algorithm online, and have done our best to make it readable, runnable and extendable.

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