Efficient unitary designs and pseudorandom unitaries from permutations (2404.16751v2)

Abstract: In this work we give an efficient construction of unitary $k$-designs using $\tilde{O}(k\cdot poly(n))$ quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over $S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that products of exponentiated sums of $S(N)$ permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-wise independent permutations, or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, we obtain the above results. The heart of our proof is a conceptual connection between the large dimension (large-$N$) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-$N$ by interpolating from the much simpler large-$N$ limit. The key technical step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$ expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension $N$.

• The paper presents a quantum algorithm that efficiently constructs approximate unitary k-designs using O(k poly(n)) gates, nearly matching lower bounds.
• The paper introduces a pseudorandom unitary ensemble by replacing random with pseudorandom permutations, ensuring computational indistinguishability under quantum-secure assumptions.
• The approach leverages large-N freeness and central limit theorem adaptations to reduce resource requirements for quantum benchmarking and cryptographic protocols.

Efficient Unitary Designs and Pseudorandom Unitaries from Permutations

The paper explores the development of efficient constructions for unitary $k$-designs and pseudorandom unitary ensembles by leveraging random permutations. The goals are twofold: first, to efficiently mimic the properties of Haar-random unitaries up to a fixed number of moments $k$ and second, to construct computationally secure pseudorandom unitaries.

Main Contributions

1. Unitary $k$-Designs Using Permutations

The authors present a method for constructing unitary $k$-designs by utilizing a quantum algorithm that maps random permutations to random unitaries. The algorithm efficiently constructs approximate unitary $k$-designs on $n$ qubits using $\tilde{O}(k \cdot \text{poly}(n))$ quantum gates. The new design is noteworthy due to its reduced dependence on $k$, closely matching known lower bounds up to logarithmic factors. This marks an improvement over prior constructions, where gate complexity scales proportionally with $nk^2$.

2. Pseudorandom Unitaries

The paper also describes an efficient mechanism for constructing a parallel-secure pseudorandom unitary ensemble. This pseudorandom unitary (PRU) construction relies on substituting pseudorandom permutations in place of random permutations to achieve computational indistinguishability from the Haar measure against nonadaptive queries. The authors demonstrate that a quantum-secure one-way function suffices to generate such PRUs, thus aligning with standard cryptographic assumptions.

Technical Insights

Large-$N$ Limit and Freeness: The authors leverage the large-$N$ limit, where distinct words formed by permutations are shown to be asymptotically free. This means distinct sequences of permutations can be treated as independent, simplifying their composition and enabling the construction of designs that closely approximate the Gaussian unitary ensemble properties.

Central Limit Theorem Adaptation: The work employs a version of the central limit theorem to ensure that sums of random phased permutations converge to a matrix with characteristics similar to a Gaussian. Such an ensemble closely resembles a Haar-random unitary in the context of nonadaptive quantum queries, facilitating the design of efficient PRUs.

Markov Inequalities and Polynomial Interpolation: The authors utilize Markov's inequality to ensure their defined functions over the inverse of the dimension $1/N$ remain bounded. This novel approach provides a means to bound finite-$N$ corrections and ensures the convergence of their ensemble towards expected random unitary properties.

Practical Implications

This work holds significant implications for quantum computing and cryptographic protocols. Efficient unitary designs can dramatically decrease the resources required for randomized benchmarking and other tasks that traditionally depended on actual Haar-random unitaries. Additionally, constructing pseudorandom unitaries that withstand quantum attacks directly from one-way functions simplifies the assumptions needed for robust quantum cryptographic protocols. Furthermore, it opens avenues for constructing efficient quantum circuits within quantum gravity studies where unitary designs play a role in addressing complexity-related paradoxes.

Future Developments

Future work could explore adaptive query security for pseudorandom unitaries, extending beyond the results for parallel-queries examined here. Moreover, the continued development of efficient implementations for permutation-based pseudorandom constructions might catalyze applications in more diverse quantum protocols. The implications for quantum systems and potentially linking them to understanding physical phenomena, such as those encountered in quantum gravity, remain an enticing domain for future investigation.