Simple constructions of linear-depth t-designs and pseudorandom unitaries (2404.12647v1)

Abstract: Uniformly random unitaries, i.e. unitaries drawn from the Haar measure, have many useful properties, but cannot be implemented efficiently. This has motivated a long line of research into random unitaries that "look" sufficiently Haar random while also being efficient to implement. Two different notions of derandomisation have emerged: $t$-designs are random unitaries that information-theoretically reproduce the first $t$ moments of the Haar measure, and pseudorandom unitaries (PRUs) are random unitaries that are computationally indistinguishable from Haar random. In this work, we take a unified approach to constructing $t$-designs and PRUs. For this, we introduce and analyse the "$PFC$ ensemble", the product of a random computational basis permutation $P$, a random binary phase operator $F$, and a random Clifford unitary $C$. We show that this ensemble reproduces exponentially high moments of the Haar measure. We can then derandomise the $PFC$ ensemble to show the following: (1) Linear-depth $t$-designs. We give the first construction of a (diamond-error) approximate $t$-design with circuit depth linear in $t$. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their $2t$-wise independent counterparts. (2) Non-adaptive PRUs. We give the first construction of PRUs with non-adaptive security, i.e. we construct unitaries that are indistinguishable from Haar random to polynomial-time distinguishers that query the unitary in parallel on an arbitary state. This follows from the $PFC$ ensemble by replacing the random phase and permutation operators with their pseudorandom counterparts. (3) Adaptive pseudorandom isometries. We show that if one considers isometries (rather than unitaries) from $n$ to $n + \omega(\log n)$ qubits, a small modification of our PRU construction achieves general adaptive security.

• The paper presents a novel PFC ensemble that simulates Haar random unitaries up to the t-th moment with diamond-error decreasing as O(t/sqrt(d)).
• The paper introduces the first construction of t-designs with circuit depth linear in t, leveraging t-wise independent approximations for random operations.
• The paper develops non-adaptively secure pseudorandom unitaries and adaptive pseudorandom isometries, paving the way for secure quantum computations and cryptographic protocols.

Analyzing Simple Constructions of Linear-Depth $t$-Designs and Pseudorandom Unitaries

The paper introduces novel constructions of linear-depth $t$-designs and pseudorandom unitaries (PRUs), driven by the need for efficient quantum computations that emulate the effects of Haar random unitaries. The work is built upon the foundations of random unitary ensembles which aim to reproduce properties of the Haar measure—a pivotal concept in quantum information theory—while being implementable with feasible resources.

Overview of Key Contributions

The authors present the "PFC ensemble," an ensemble composed of a product of three components: a random permutation operator $P$, a random binary phase operator $F$, and a random Clifford unitary $C$. The ensemble is shown to be a diamond-error $t$-design with a vanishingly small error for $t$ up to superpolynomial in the number of qubits. This asserts that the $PFC$ ensemble behaves indistinguishably from Haar random unitaries up to the $t$-th moment, although with an error decreasing as $\mathcal{O}(t/\sqrt{d})$ for dimension $d=2^n$.

Efficient $t$-Design Constructions

Leveraging $t$-wise independence principles from classical pseudo-randomness, the authors present the first construction of $t$-designs that feature a circuit depth linear in $t$. They achieve this by replacing fully random functions and permutations with their respective $t$-wise independent counterparts. The result is a diamond-error approximate $t$-design with circuit size and depth scaling as $O(t \cdot \mathrm{poly}(n))$. This offers significant improvements in practicality, offering a pathway to efficiently simulate high-moment Haar randomness.

Pseudorandom Unitaries with Non-Adaptive Security

A parallel contribution focuses on constructing non-adaptively secure pseudorandom unitaries. In classical terms, a pseudorandom function appears random to any polynomial-time bounded observer. The paper extends this concept to quantum computing by utilizing pseudorandom permutations and functions, ensuring that polynomial-time quantum adversaries cannot discern these from true Haar random unitaries under non-adaptive access conditions.

Adaptive Security and Pseudorandom Isometries

While adaptive security—protection against adversaries which adaptively query the system—was not achieved for PRUs, a related construct known as pseudorandom isometries (PRIs) obtains such a security level. By transforming the input space slightly, from $n$ qubits to $n + \omega(\log n)$ qubits, adaptive security is demonstrated. This represents a promising avenue for applications requiring secure quantum state generation.

Implications and Future Directions

The paper's advancements open numerous possibilities in both theoretical and practical domains of quantum computing. Efficient $t$-designs have significant implications for simulating quantum dynamics, such as quantum chaos, and optimizing random quantum circuits. Non-adaptive PRUs might find use in cryptographic protocols where the efficiency of the quantum process is paramount, while adaptive PRIs suggest new avenues in secure multiparty quantum cryptography.

While this research answers several open questions about the efficiency of $t$-designs and the construction of PRUs, further exploration could enhance their security to fully adaptive paradigms. Additionally, extending the computational indistinguishability to scenarios allowing inverse or controlled access to the unitary could broaden applications in complexity theory and cryptographic primitives.

In summary, the approach presented in the paper provides crucial insights and practical methods for simulating Haar measure properties with quantum computational efficiency, bridging gaps in longstanding theoretical and practical endeavors in quantum information science.