Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x> to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with O(N) XOR queries but was conjectured to require Ω(N) in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using O(N) in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and Θ(N) in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.
This paper presents a novel approach to optimizing the linear layer of block ciphers using the matrix decomposition framework. It is observed that the reduction properties proposed by Xiang et al. (in FSE 2020) need to be improved. To address these limitations, we propose a new reduction framework with a complete reduction algorithm and swapping algorithm. Our approach formulates matrix decomposition as a new framework with an adaptive objective function and converts the problem to a Graph Isomorphism problem (GI problem). Using the new reduction algorithm, we were able to achieve lower XOR counts and depths of quantum implementations under the s-XOR metric. Our results outperform previous works for many linear layers of block ciphers and hash functions; some of them are better than the current g-XOR implementation. For the AES MixColumn operation, we get two implementations with 91 XOR counts and depth 13 of in-place quantum implementation, respectively.
We consider the power of local algorithms for approximately solving Max k XOR, a generalization of two constraint satisfaction problems previously studied with classical and quantum algorithms (MaxCut and Max E3LIN2). In Max k XOR each constraint is the XOR of exactly k variables and a parity bit. On instances with either random signs (parities) or no overlapping clauses and D+1 clauses per variable, we calculate the expected satisfying fraction of the depth-1 QAOA from Farhi et al [arXiv:1411.4028] and compare with a generalization of the local threshold algorithm from Hirvonen et al [arXiv:1402.2543]. Notably, the quantum algorithm outperforms the threshold algorithm for k > 4 .On the other hand, we highlight potential difficulties for the QAOA to achieve computational quantum advantage on this problem. We first compute a tight upper bound on the maximum satisfying fraction of nearly all large random regular Max k XOR instances by numerically calculating the ground state energy density P(k) of a mean-field k -spin glass [arXiv:1606.02365]. The upper bound grows with k much faster than the performance of both one-local algorithms. We also identify a new obstruction result for low-depth quantum circuits (including the QAOA) when k=3 , generalizing a result of Bravyi et al [arXiv:1910.08980] when k=2 . We conjecture that a similar obstruction exists for all k .