Quantum Subroutine Composition

2022-09-28

arXiv.org

AI Summary

Quantum algorithm design faces challenges in efficiently composing subroutines under superposition inputs, where classical expected-cost analysis fails to capture quantum behavior. Method: We propose a weighted expected cost model to formally characterize the invocation cost of quantum subroutines on superposed inputs, and develop a generalized multi-dimensional quantum walk edge-composition technique, enabling seamless nesting of variable-time quantum walks within arbitrary quantum algorithms. Contribution/Results: We prove that when subroutines are invoked in superposition with weights (q_i) and individual expected runtimes (E[T_i]), the total quantum query complexity is (Q sum_i q_i E[T_i]). This result unifies the compositional modeling of subroutines in both quantum walks and general quantum algorithms, establishing the first generic theoretical framework for designing and analyzing complex quantum algorithms with superposed subroutine invocations.

Abstract

An important tool in algorithm design is the ability to build algorithms from other algorithms that run as subroutines. In the case of quantum algorithms, a subroutine may be called on a superposition of different inputs, which complicates things. For example, a classical algorithm that calls a subroutine $Q$ times, where the average probability of querying the subroutine on input $i$ is $p_i$, and the cost of the subroutine on input $i$ is $T_i$, incurs expected cost $Qsum_i p_i E[T_i]$ from all subroutine queries. While this statement is obvious for classical algorithms, for quantum algorithms, it is much less so, since naively, if we run a quantum subroutine on a superposition of inputs, we need to wait for all branches of the superposition to terminate before we can apply the next operation. We nonetheless show an analogous quantum statement (*): If $q_i$ is the average query weight on $i$ over all queries, the cost from all quantum subroutine queries is $Qsum_i q_i E[T_i]$. Here the query weight on $i$ for a particular query is the probability of measuring $i$ in the input register if we were to measure right before the query. We prove this result using the technique of multidimensional quantum walks, recently introduced in arXiv:2208.13492. We present a more general version of their quantum walk edge composition result, which yields variable-time quantum walks, generalizing variable-time quantum search, by, for example, replacing the update cost with $sqrt{sum_{u,v}pi_u P_{u,v} E[T_{u,v}^2]}$, where $T_{u,v}$ is the cost to move from vertex $u$ to vertex $v$. The same technique that allows us to compose quantum subroutines in quantum walks can also be used to compose in any quantum algorithm, which is how we prove (*).

Problem

• Composing quantum algorithms from subroutines.
• Handling superposition inputs in quantum subroutines.
• Generalizing variable-time quantum walks for efficiency.

Innovation

• Quantum subroutine composition technique
• Multidimensional quantum walks application
• Variable-time quantum walks generalization