Composability of global phase invariant distance and its application to approximation error management

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Abstract
Many quantum algorithms can be written as a composition of unitaries, some of which can be exactly synthesized by a universal fault-tolerant gate set like Clifford+T, while others can be approximately synthesized. One task of a quantum compiler is to synthesize each approximately synthesizable unitary up to some approximation error, such that the error of the overall unitary remains bounded by a certain amount. In this paper we consider the case when the errors are measured in the global phase invariant distance. Apart from deriving a relation between this distance and the Frobenius norm, we show that this distance composes. If a unitary is written as a composition (product and tensor product) of other unitaries, we derive bounds on the error of the overall unitary as a function of the errors of the composed unitaries. Our bound is better than the sum-of-error bound, derived by Bernstein- Vazirani(1997), for the operator norm. This builds the intuition that working with the global phase invariant distance might give us a lower resource count while synthesizing quantum circuits. Next we consider the following problem. Suppose we are given a decomposition of a unitary, that is, the unitary is expressed as a composition of other unitaries. We want to distribute the errors in each component such that the resource-count (specifically T-count) is optimized. We consider the specific case when the unitary can be decomposed such that the R z (θ) gates are the only approximately synthesizable component. We prove analytically that for both the operator norm and global phase invariant distance, the error should be distributed equally among these components (given some approximations). The optimal number of T-gates obtained by using the global phase invariant distance is less than what is obtained using the operator norm. Furthermore, we show that in case of approximate Quantum Fourier Transform, the error obtained by pruning rotation gates is less when measured in this distance, rather than the operator norm.
Funding Information
  • NTT Research
  • Government of Canada

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