lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle}of Hoyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

DOI: 10.26421/qic22.7-8-3

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Variations on Quantum Adversary

Aleksandrs Belovs

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\Omega(\sqrt{n})

Ω(\sqrt{n})

lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle of Høyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

The Index Erasure problem asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight

\Omega(\sqrt{n})

We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a

T

-query adversary with

S

qubits of non-uniform advice can recover a random key

k

from the

n

-qubit binary phase state

|ψ_k\rangle \propto \sum_{x} R(k,x) |x\rangle

with probability at most

O(\frac{T^2 + \sqrt{ST}}{N})

for

N=2^n

. In contrast, the best known bound for post-quantum one-way functions is

O(\frac{T^2 + ST}{N})

, with a trivial attack at

S = N

. This demonstrates a new advantage of quantum cryptography over classical cryptography:

n

qubits of communication suffice for security against preprocessing attacks with space up to

N^2

rather than

N

. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of

O(\frac{T^2}N + \sqrt{\frac{ST}N})

. (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight

O(\frac{\sqrt{S}}N)

bound for the pseudorandomness of random binary phase states against space

S

distinguishers.