Strong random unitaries and fast scrambling

TL;DR

The paper tackles how fast quantum systems can mimic Haar-random dynamics under all relevant queries, defining strong unitary designs and strong PRUs that are indistinguishable from Haar when querying $U$, $U^ $, $U^*$, or $U^T$. It introduces the LRFC ensemble and a gluing framework to achieve optimal $ ext{depth} = O( ext{log} obreakspace n)$ for strong designs and PRUs on all-to-all connected networks, with ancilla-free variants reachable in polylogarithmic depth under cryptographic assumptions such as subexponential LWE. It further shows that all-to-all random circuits of Haar-random two-qubit gates yield strong unitary designs in depth $O( ext{log}^2 n)$, and develops ancilla-free constructions for PRUs, broadening the physical relevance of pseudorandomness in quantum dynamics. By linking these constructions to fast scrambling, the work provides an operational proof that the fastest scrambling quantum systems reproduce Haar-random behavior at timescales proportional to $ ext{log} obreakspace n$, with strong diagnostics including OTOCs, Hayden–Preskill decoding, and operator-size growth aligning with Haar values at logarithmic times. Overall, the paper advances both quantum information theory and black-hole-inspired physics by delivering optimal-depth, ancilla-agnostic certifications of scrambling and cryptographic robustness against a wide class of unitary queries.

Abstract

Understanding how fast physical systems can resemble Haar-random unitaries is a fundamental question in physics. Many experiments of interest in quantum gravity and many-body physics, including the butterfly effect in quantum information scrambling and the Hayden-Preskill thought experiment, involve queries to a random unitary $U$ alongside its inverse $U^\dagger$, conjugate $U^*$, and transpose $U^T$. However, conventional notions of approximate unitary designs and pseudorandom unitaries (PRUs) fail to capture these experiments. In this work, we introduce and construct strong unitary designs and strong PRUs that remain robust under all such queries. Our constructions achieve the optimal circuit depth of $O(\log n)$ for systems of $n$ qubits. We further show that strong unitary designs can form in circuit depth $O(\log^2 n)$ in circuits composed of independent two-qubit Haar-random gates, and that strong PRUs can form in circuit depth $\text{poly}(\log n)$ in circuits with no ancilla qubits. Our results provide an operational proof of the fast scrambling conjecture from black hole physics: every observable feature of the fastest scrambling quantum systems reproduces Haar-random behavior at logarithmic times.

Figures (5)

• Figure 1: Illustration of our main results. (a) A strong approximate unitary $k$-design is a random unitary ensemble that is indistinguishable from Haar in any quantum experiment that queries $U$ or its inverse (i.e. time-reversal), conjugate, or transpose $k$ times. A strong pseudorandom unitary (PRU) is similarly indistinguishable in any polynomial-time experiment. (b) We construct strong unitary designs and PRUs on $n$ qubits in depth $\mathcal{O}(\log n)$. Our constructions use long-range two-qubit gates to scramble quantum information over all $n$ qubits as fast as possible. (c) In comparison, low-depth one-dimensional quantum circuits can only scramble information over local regions. This allows them to form conventional designs schuster2024randomlaracuente2024approximate and PRUs schuster2024random, but not strong designs or strong PRUs.
• Figure 2: Our constructions of strong unitary $k$-designs and strong PRUs. (a) The Luby-Rackoff-Function-Clifford (LRFC) ensemble sandwiches classical shuffle and phase gates (pink and orange) between random Clifford unitaries (green). It forms a strong unitary design and a strong PRU in the stated circuit depths. (b) To further reduce these depths, we consider a glued construction, with two layers of small $2\xi$-qubit random unitaries (various colors) sandwiched between strong $n$-qubit unitary 2-designs (green). This forms a strong unitary $k$-design when $\xi = \Omega(\log (nk/\varepsilon))$ and a strong PRU when $\xi = \omega(\log n)$. We instantiate each small unitary with the LRFC ensemble. (c) To obtain ancilla-free constructions, we replace each $n$-qubit 2-design with a fast scrambling circuit of depth $\log n$ composed of $2\xi$-qubit 2-designs. For ancilla-free strong unitary designs consisting of Haar-random two-qubit gates, each small unitary is drawn from a random circuit on $2\xi$ qubits. For ancilla-free strong PRUs, each small unitary is implemented by reusing neighboring qubits as ancillae.
• Figure 3: Illustration of key ideas from the proof of Theorem \ref{['thm:strong-design-depth']} and Theorem \ref{['thm:strong-PRU-depth']}. (a) To analyze our glued construction, we prove that two strong unitary $k$-designs "glue" together whenever they are sandwiched by larger unitary 2-designs. This does not hold in the absence of the larger 2-designs. (b) To show that the blocked fast scrambling circuit forms a strong 2-design, we prove that any unitary ensemble that uniformly spreads quantum information (purple) is a strong 2-design. The circuit spreads information over $n$ qubits in $\log n$ layers. (c) To replace each $2\xi$-qubit unitary with a local random circuit, we prove a technical lemma that translates spectral gaps brandao2016localhaferkamp2022randomchen2024incompressibility to strong unitary designs.
• Figure 4: By allowing queries to $U^\dagger$ and $U^*$, strong random unitaries capture hallmark features of quantum information scrambling. (a) In a strong unitary $k$-design, every $k$-point time-ordered and out-of-time-order correlation function decays to near zero with high probability. (b) Strong random unitaries are good encoders of quantum information, as in the Hayden-Preskill thought experiment hayden2007black. This follows because the decoding protocol uses the conjugate random unitary yoshida2017efficient. (c) In a strong unitary 4-design or strong pseudorandom unitary, the operator size distribution approaches its Haar-random value to within small total variational distance.
• Figure 5: Reformulation of any quantum experiment that makes any $k$ queries to $U$, $U^*$, $U^T$, or $U^\dagger$ as an alternative experiment that makes a single query to $U^{\otimes p} \otimes U^{*,\otimes q}$ and performs $k$ post-selections on $n$-qubit Bell states. Here, $p$ counts the number of applications of $U$ and $U^T$ and $q$ counts the number of applications of $U^*$ and $U^\dagger$. The subsystem labels match the notation used in our proofs based on the path-recording framework (Appendix \ref{['sec: proof strong gluing']} and \ref{['app: LRFC']}).

Theorems & Definitions (146)

• Conjecture 1: Fast scrambling conjecture, Sekino and Susskind sekino2008fast
• Conjecture 2: Operational fast scrambling conjecture
• Conjecture 3: Strong fast scrambling conjecture
• Theorem 1: Fast formation of strong unitary designs
• Proposition 1
• Theorem 2: Fast formation of strong PRUs
• Theorem 3: The LRFC ensemble is a strong unitary design
• Theorem 4: The LRFC ensemble is a strong PRU
• Theorem 5: The scrambled two-layer ensemble is a strong unitary design
• Theorem 6: The scrambled two-layer ensemble is a strong PRU