Simple constructions of linear-depth t-designs and pseudorandom unitaries

• URL: http://arxiv.org/abs/2404.12647v1
• Date: Fri, 19 Apr 2024 06:13:02 GMT
• Title: Simple constructions of linear-depth t-designs and pseudorandom unitaries
• Authors: Tony Metger, Alexander Poremba, Makrand Sinha, Henry Yuen,
• Abstract summary: Uniformly random unitaries, i.e. unitaries drawn from the Haar measure, have many useful properties, but cannot be implemented efficiently. Two different notions of derandomisation have emerged: $t-designs are random unitaries that reproduce the first $t moments of the Haar measure, and pseudorandom unitaries (PRUs) are random unitaries that are computationally indistinguishable from Haar random.
• Score: 40.72668922727092
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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