DeepAI

# (Pseudo) Random Quantum States with Binary Phase

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.

• 13 publications
• 4 publications
06/30/2022

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### Cayley path and quantum computational supremacy: A proof of average-case #P-hardness of Random Circuit Sampling with quantified robustness

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### K-sparse Pure State Tomography with Phase Estimation

Quantum state tomography (QST) for reconstructing pure states requires e...
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### Scalable Pseudorandom Quantum States

Efficiently sampling a quantum state that is hard to distinguish from a ...
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### Computational pseudorandomness, the wormhole growth paradox, and constraints on the AdS/CFT duality

A fundamental issue in the AdS/CFT correspondence is the wormhole growth...
06/28/2022

### Improved resource-tunable near-term quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra

Transition amplitudes and transition probabilities are relevant to many ...