How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$ . We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$ , in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.

cond-mat.stat-mechCross-listedarXiv:2609.13387Sep 15, 20260 citations

Discrete-Time Quantum Walk with Memory on the Cayley Graph of the Dihedral Group

Wenjing Dai, Jiabin Yuan, Dan Li

International Journal of Theoretical PhysicsDec 10, 20195 citations

Continuous-Time Quantum Walks on the Symmetric Group

Heath Gerhardt, John Watrous

Lecture notes in computer scienceJan 1, 200374 citations

Single-shuffle card-based protocol with eight cards per gate and its extensions

Kazunari Tozawa, Hiraku Morita, Takaaki Mizuki

Abstract Card-based cryptography allows us to securely compute arbitrary functions using a deck of physical cards. Its performance is mainly measured by the number of used cards and shuffles, and there is a line of work that aims to reduce either of them. One seminal work is the card-based garbled circuit technique by Shinagawa and Nuida (Discret Appl Math 289:248–261, 2021, https://doi.org/10.1016/j.dam.2020.10.013 ), which allows the construction of a card-based protocol for any Boolean function with a single shuffle. Their construction requires $2n + 24g$ 2 n + 24 g cards for an n-input Boolean function that is represented by g logical gates. In this paper, we reduce the number of cards to $2n + 8g$ 2 n + 8 g for arbitrary functions while keeping it working with only one shuffle. In addition, we propose two types of extensions to support numerical encoding and multi-input gates. In the extended scheme, the free-ADD technique, obtained by generalizing the free-XOR technique by Manabe and Shinagawa (Deng J, Kolesnikov V, Schwarzmann AA (eds) CANS 2023, LNCS, vol 14342. Springer, Singapore, pp 232–248, 2023, https://doi.org/10.1007/978-981-99-7563-1-11 ), is available. The free-ADD technique allows our scheme to evaluate any n-input symmetric Boolean function using $2n^2+6n+2$ 2 n 2 + 6 n + 2 cards.Abstract Card-based cryptography allows us to securely compute arbitrary functions using a deck of physical cards. Its performance is mainly measured by the number of used cards and shuffles, and there is a line of work that aims to reduce either of them. One seminal work is the card-based garbled circuit technique by Shinagawa and Nuida (Discret Appl Math 289:248–261, 2021, https://doi.org/10.1016/j.dam.2020.10.013 ), which allows the construction of a card-based protocol for any Boolean function with a single shuffle. Their construction requires $$2n + 24g$$ 2 n + 24 g cards for an n-input Boolean function that is represented by g logical gates. In this paper, we reduce the number of cards to $$2n + 8g$$ 2 n + 8 g for arbitrary functions while keeping it working with only one shuffle. In addition, we propose two types of extensions to support numerical encoding and multi-input gates. In the extended scheme, the free-ADD technique, obtained by generalizing the free-XOR technique by Manabe and Shinagawa (Deng J, Kolesnikov V, Schwarzmann AA (eds) CANS 2023, LNCS, vol 14342. Springer, Singapore, pp 232–248, 2023, https://doi.org/10.1007/978-981-99-7563-1-11 ), is available. The free-ADD technique allows our scheme to evaluate any n-input symmetric Boolean function using $$2n^2+6n+2$$ 2 n 2 + 6 n + 2 cards.

Natural ComputingJan 9, 20253 citations

S_n

. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of

S_n

, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.

2n + 24g

2 n + 24 g cards for an n-input Boolean function that is represented by g logical gates. In this paper, we reduce the number of cards to

2n + 8g

2 n + 8 g for arbitrary functions while keeping it working with only one shuffle. In addition, we propose two types of extensions to support numerical encoding and multi-input gates. In the extended scheme, the free-ADD technique, obtained by generalizing the free-XOR technique by Manabe and Shinagawa (Deng J, Kolesnikov V, Schwarzmann AA (eds) CANS 2023, LNCS, vol 14342. Springer, Singapore, pp 232–248, 2023, https://doi.org/10.1007/978-981-99-7563-1-11 ), is available. The free-ADD technique allows our scheme to evaluate any n-input symmetric Boolean function using

2n^2+6n+2

2 n 2 + 6 n + 2 cards.