(Pseudo) Random Quantum States with Binary Phase

by   Zvika Brakerski, et al.

We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random 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 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 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.


page 1

page 2

page 3

page 4


Multivariate trace estimation in constant quantum depth

There is a folkloric belief that a depth-Θ(m) quantum circuit is needed ...

Automatically Differentiable Quantum Circuit for Many-qubit State Preparation

Constructing quantum circuits for efficient state preparation belongs to...

SWAP Test for an Arbitrary Number of Quantum States

We develop a recursive algorithm to generalize the quantum SWAP test for...

Scalable Pseudorandom Quantum States

Efficiently sampling a quantum state that is hard to distinguish from a ...

K-sparse Pure State Tomography with Phase Estimation

Quantum state tomography (QST) for reconstructing pure states requires e...

Quantum Pseudorandom Scramblers

Quantum pseudorandom state generators (PRSGs) have stimulated exciting d...

Please sign up or login with your details

Forgot password? Click here to reset