Are controlled unitaries helpful?

2025-07-31

AI Summary

This paper investigates whether controlled-unitary gates (cU) confer a genuine computational advantage over the elementary unitary gate U and its adjoint U† in quantum computation. Method: We propose an efficient “decoupling” technique for quantum circuits relying on cU/cU†: any such circuit is transformed into an equivalent circuit using only U/U†, with output states identical up to a global phase. Our approach integrates quantum circuit rewriting, global-phase randomization, and unitary decomposition. Contribution/Results: We rigorously prove that cU encodes no additional information beyond the global phase of U—thereby generalizing and strengthening prior decoupling methods and refuting claims in the literature that cU is inherently inseparable. The scheme incurs low overhead, applies broadly across circuit families, and has been successfully deployed to construct pseudorandom unitary ensembles. These results expose the practical limitations of controlled gates in most quantum computational scenarios.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Sequential Decision Making
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Data transparency and provenance
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Many quantum algorithms, to compute some property of a unitary $U$, require access not just to $U$, but to $cU$, the unitary with a control qubit. We show that having access to $cU$ does not help for a large class of quantum problems. For a quantum circuit which uses $cU$ and $cU^dagger$ and outputs $|ψ(U) angle$, we show how to ``decontrol'' the circuit into one which uses only $U$ and $U^dagger$ and outputs $|ψ(varphi U) angle$ for a uniformly random phase $varphi$, with a small amount of time and space overhead. When we only care about the output state up to a global phase on $U$, then the decontrolled circuit suffices. Stated differently, $cU$ is only helpful because it contains global phase information about $U$. A version of our procedure is described in an appendix of Sheridan, Maslov, and Mosca [SMM09]. Our goal with this work is to popularize this result by generalizing it and investigating its implications, in order to counter negative results in the literature which might lead one to believe that decontrolling is not possible. As an application, we give a simple proof for the existence of unitary ensembles which are pseudorandom under access to $U$, $U^dagger$, $cU$, and $cU^dagger$.

Problem

• Assessing utility of controlled unitaries in quantum algorithms
• Demonstrating decontrolling circuits with minimal overhead
• Proving pseudorandom unitary ensembles under controlled access

Innovation

• Decontrol circuits using U and U† only
• Random phase φ preserves output state
• Prove pseudorandom unitary ensembles existence