# (Pseudo) Random Quantum States with Binary Phase

## TL;DR

This paper proves that binary phase superpositions are indistinguishable from Haar random states, enabling explicit constructions of pseudorandom quantum states and t-designs with efficient, simple quantum circuits, including real-valued states.

## Contribution

It provides a proof of a conjecture relating binary phase states to Haar states and introduces efficient, explicit constructions of pseudorandom states and t-designs with simple quantum circuits.

## Key findings

- Binary phase superpositions are statistically indistinguishable from Haar random states.
- Explicit constructions of pseudorandom quantum states from post-quantum pseudorandom functions.
- Efficient quantum circuit designs for t-designs with polylogarithmic depth and real amplitudes.

## Abstract

We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random \emph{binary} phase is statistically indistinguishable from a Haar random state. That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state. As a consequence, we get a provable elementary construction of \emph{pseudorandom} quantum states from post-quantum pseudorandom functions. Generating pseduorandom quantum states is desirable for physical applications as well as for computational tasks such as quantum money. We observe that replacing the pseudorandom function with a $(2t)$-wise independent function (either in our construction or in previous work), results in an explicit construction for \emph{quantum state $t$-designs} for all $t$. In fact, we show that the circuit complexity (in terms of both circuit size and depth) of constructing $t$-designs is bounded by that of $(2t)$-wise independent functions. Explicitly, while in prior literature $t$-designs required linear depth (for $t > 2$), this observation shows that polylogarithmic depth suffices for all $t$. We note that our constructions yield pseudorandom states and state designs with only real-valued amplitudes, which was not previously known. Furthermore, generating these states require quantum circuit of restricted form: applying one layer of Hadamard gates, followed by a sequence of Toffoli gates. This structure may be useful for efficiency and simplicity of implementation.