(Pseudo) Random Quantum States with Binary Phase

Abstract

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 realvalued 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.

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“…In fact, we can simplify the construction. It was shown in[32] that the same family of states is still pseudorandom if we replace the root of unity ωN by −1.…”

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“…Pseudorandom quantum states [31][32][33] are sets of quantum states that can be efficiently generated but computationally indistinguishable from Haar random states. (The formal definition is given in Definition 1.)…”

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“…Alternatively, quantum t-design circuits are a popular way to generate pseudo-random states [38,39,40]. A t-design circuit outputs a state that is indistinguishable from states drawn from a random Haar measure.…”

“…These t-designs in a quantum computer are equivalent to t-wise independent vectors in the classical world [38]. Short-depth circuits exist (though not as short as above) that are approximate t-designs [40]. Such t-design circuits can be used to generate the random states |v l for trace estimation.…”