TY - GEN

T1 - (Pseudo) Random Quantum States with Binary Phase

AU - Brakerski, Zvika

AU - Shmueli, Omri

N1 - Publisher Copyright:
© 2019, International Association for Cryptologic Research.

PY - 2019

Y1 - 2019

N2 - 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 pseudorandom 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.

AB - 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 pseudorandom 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.

UR - http://www.scopus.com/inward/record.url?scp=85076966177&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-36030-6_10

DO - 10.1007/978-3-030-36030-6_10

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AN - SCOPUS:85076966177

SN - 9783030360290

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 229

EP - 250

BT - Theory of Cryptography - 17th International Conference, TCC 2019, Proceedings

A2 - Hofheinz, Dennis

A2 - Rosen, Alon

PB - Springer

T2 - 17th International Conference on Theory of Cryptography, TCC 2019

Y2 - 1 December 2019 through 5 December 2019

ER -