Haar-random unitaries provide a canonical model of generic random quantum evolution, but they typically have exponential description and circuit complexity. Although pseudorandom unitaries efficiently emulate a single Haar-random unitary, many tasks require an entire reusable family of independently random-looking operations. We introduce pseudorandom function-like unitaries (PRFU), which generate such a family indexed by public labels using only a single short key. We distinguish classical- and coherent-label access and establish security under adaptive quantum queries. For classical labels, we provide a generic construction from a post-quantum pseudorandom function and a pseudorandom unitary. For coherent labels, we use an indexed path-recording framework to analyze interference across labels and prove security of a construction based on quantum-secure function and permutation primitives. We further show that PRFUs generate pseudorandom channels and quantum combs with private memory, which are secure against adaptive interventions and concurrent sessions. Combining PRFUs with unitary gluing yields a one-key family of pseudorandom unitaries whose supported register widths can be chosen after key generation. Further applications include nonce-resolved quantum authentication, coherently masked QRAM queries, and efficient emulation of random multi-time dynamics. Our results extend quantum pseudorandomness from individual operations to efficiently generated families of random-looking quantum dynamics.Haar-random unitaries provide a canonical model of generic random quantum evolution, but they typically have exponential description and circuit complexity. Although pseudorandom unitaries efficiently emulate a single Haar-random unitary, many tasks require an entire reusable family of independently random-looking operations. We introduce pseudorandom function-like unitaries (PRFU), which generate such a family indexed by public labels using only a single short key. We distinguish classical- and coherent-label access and establish security under adaptive quantum queries. For classical labels, we provide a generic construction from a post-quantum pseudorandom function and a pseudorandom unitary. For coherent labels, we use an indexed path-recording framework to analyze interference across labels and prove security of a construction based on quantum-secure function and permutation primitives. We further show that PRFUs generate pseudorandom channels and quantum combs with private memory, which are secure against adaptive interventions and concurrent sessions. Combining PRFUs with unitary gluing yields a one-key family of pseudorandom unitaries whose supported register widths can be chosen after key generation. Further applications include nonce-resolved quantum authentication, coherently masked QRAM queries, and efficient emulation of random multi-time dynamics. Our results extend quantum pseudorandomness from individual operations to efficiently generated families of random-looking quantum dynamics.

quant-pharXiv:2610.01038Oct 2, 20260 citations

Random Processes

Kedar Khare, Mansi Butola, Sunaina Rajora

Springer eBooksJan 1, 20230 citations

On the pseudorandomness of simple quantum processes

Jesko Dujmovic, Jonas Haferkamp, Alexander Poremba

Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed $t$ , we give an efficiently samplable family $\{ν_n\}_n$ of distributions on one- and two-qubit gates such that, after $T=O_t(n^2\log^2 n)$ independent steps, the resulting $n$ -qubit ensemble is an approximate unitary $t$ -design with negligible error $\exp(-Ω(\log^2 n))$ , yet an efficient quantum algorithm distinguishes it from random using only $O_t(\log^2 n)$ queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed $t$, we give an efficiently samplable family $\{ν_n\}_n$ of distributions on one- and two-qubit gates such that, after $T=O_t(n^2\log^2 n)$ independent steps, the resulting $n$-qubit ensemble is an approximate unitary $t$-design with negligible error $\exp(-Ω(\log^2 n))$, yet an efficient quantum algorithm distinguishes it from random using only $O_t(\log^2 n)$ queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.

quant-pharXiv:2610.02100Oct 2, 20260 citations

Random Versus Pseudorandom Physics

Joseph Edward Brierly

Physical processes can be compared to a lottery. This article compares natural physical events to artificial ones created by man’s intervention. Manmade physical processes like a lottery cannot be perfectly random. Pure random behavior can only occur in nature without manmade intervention. Said another way all manmade lotteries have a nonrandom bias. The best that can happen is to contrive a lottery that converges on pure random behavior. This article explains why manual intervention invariably results in pseudorandom bias. In this article the label pseudorandom applies to any process occurring in nature that requires man’s intervention. The article asserts that Repulsion Gravity can be viewed as the element in nature that causes true randomness. Physical processes like Brownian Motion represent pure random behavior. This article compares random to pseudorandom behavior.Physical processes can be compared to a lottery. This article compares natural physical events to artificial ones created by man’s intervention. Manmade physical processes like a lottery cannot be perfectly random. Pure random behavior can only occur in nature without manmade intervention. Said another way all manmade lotteries have a nonrandom bias. The best that can happen is to contrive a lottery that converges on pure random behavior. This article explains why manual intervention invariably results in pseudorandom bias. In this article the label pseudorandom applies to any process occurring in nature that requires man’s intervention. The article asserts that Repulsion Gravity can be viewed as the element in nature that causes true randomness. Physical processes like Brownian Motion represent pure random behavior. This article compares random to pseudorandom behavior.

Journal of Physics & Optics SciencesJul 1, 20230 citations

t

, we give an efficiently samplable family

\{ν_n\}_n

of distributions on one- and two-qubit gates such that, after

T=O_t(n^2\log^2 n)

independent steps, the resulting

n

-qubit ensemble is an approximate unitary

t

-design with negligible error

\exp(-Ω(\log^2 n))

, yet an efficient quantum algorithm distinguishes it from random using only

O_t(\log^2 n)

queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.