Gábor Ivanyos

Jonathan Allcock, Joao F. Doriguello, Gábor Ivanyos et al.

Bell sampling is a simple yet powerful tool based on measuring two copies of a quantum state in the Bell basis, and has found applications in a plethora of problems related to stabiliser states and measures of magic. However, it was not known how to generalise the procedure from qubits to $d$-level systems -- qudits -- for all dimensions $d > 2$ in a useful way. Indeed, a prior work of the authors (arXiv'24) showed that the natural extension of Bell sampling to arbitrary dimensions fails to provide meaningful information about the quantum states being measured. In this paper, we overcome the difficulties encountered in previous works and develop a useful generalisation of Bell sampling to qudits of all $d\geq 2$. At the heart of our primitive is a new unitary, based on Lagrange's four-square theorem, that maps four copies of any stabiliser state $|\mathcal{S}\rangle$ to four copies of its complex conjugate $|\mathcal{S}^\ast\rangle$ (up to some Pauli operator), which may be of independent interest. We then demonstrate the utility of our new Bell sampling technique by lifting several known results from qubits to qudits for any $d\geq 2$: 1. Learning stabiliser states in $O(n^3)$ time with $O(n)$ samples; 2. Solving the Hidden Stabiliser Group Problem in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 3. Testing whether $|ψ\rangle$ has stabiliser size at least $d^t$ or is $\varepsilon$-far from all such states in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 4. Clifford circuits with at most $n/2$ single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether $|ψ\rangle$ has stabiliser fidelity at least $1-\varepsilon_1$ or at most $1-\varepsilon_2$ with $O(d^2/\varepsilon_2)$ samples if $\varepsilon_1 = 0$ or $O(d^2/\varepsilon_2^2)$ samples if $\varepsilon_1 = O(d^{-2})$.

Gábor Ivanyos, Youming Qiao

We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices $(B_1, \dots, B_m)$ and $(C_1, \dots, C_m)$, whether there exists an invertible matrix $A$ such that for every $i\in\{1, \dots, m\}$, $A^tB_iA=C_i$. We show that this problem can be solved in randomized polynomial time over finite fields of odd size, the real field, and the complex field. The second problem asks to decide, given a tuple of square matrices $(B_1, \dots, B_m)$, whether there exist invertible matrices $A$ and $D$, such that for every $i\in\{1, \dots, m\}$, $AB_iD$ is (skew-)symmetric. We show that this problem can be solved in deterministic polynomial time over fields of characteristic not $2$. For both problems we exploit the structure of the underlying $*$-algebras, and utilize results and methods from the module isomorphism problem. Applications of our results range from multivariate cryptography, group isomorphism, to polynomial identity testing. Specifically, these results imply efficient algorithms for the following problems. (1) Test isomorphism of quadratic forms with one secret over a finite field of odd size. This problem belongs to a family of problems that serves as the security basis of certain authentication schemes proposed by Patarin (Eurocrypto 1996). (2) Test isomorphism of $p$-groups of class 2 and exponent $p$ ($p$ odd) with order $p^k$ in time polynomial in the group order, when the commutator subgroup is of order $p^{O(\sqrt{k})}$. (3) Deterministically reveal two families of singularity witnesses caused by the skew-symmetric structure, which represents a natural next step for the polynomial identity testing problem following the direction set up by the recent resolution of the non-commutative rank problem (Garg et al., FOCS 2016; Ivanyos et al., ITCS 2017).

Andrew M. Childs, Gábor Ivanyos

We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and discrete log as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete log problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete log problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to $k \ge 2$ generators in a black-box abelian semigroup of order $N$ requires $\tilde Θ(N^{\frac{1}{2}-\frac{1}{2k}})$ quantum queries.