We study a generalization of entanglement testing which we call the “hidden cut problem.” Taking as input copies of an n-qubit pure state which is product across an unknown bipartition, the goal is to learn precisely where the state is unentangled, i.e. to determine which of the exponentially many possible cuts separates the state. We give a polynomial-time quantum algorithm which can find the cut using O(n/є2) many copies of the state, which is optimal up to logarithmic factors. Our algorithm also generalizes to learn the entanglement structure of arbitrary product states. In the special case of Haar-random states, we further show that our algorithm requires circuits of only constant depth. To develop our algorithm, we introduce a state generalization of the hidden subgroup problem (StateHSP) which might be of independent interest, in which one is given a quantum state invariant under an unknown subgroup action, with the goal of learning the hidden symmetry subgroup. We show how the hidden cut problem can be formulated as a StateHSP with a carefully chosen Abelian group action. We then prove that Fourier sampling on the hidden cut state produces similar outcomes as a variant of the well-known Simon’s problem, allowing us to find the hidden cut efficiently. Therefore, our algorithm can be interpreted as an extension of Simon’s algorithm to entanglement testing. We discuss possible applications of StateHSP and hidden cut problems to cryptography and pseudorandomness.
We study a generalization of entanglement testing which we call the “hidden cut problem.” Taking as input copies of an n-qubit pure state which is product across an unknown bipartition, the goal is to learn precisely where the state is unentangled, i.e. to determine which of the exponentially many possible cuts separates the state. We give a polynomial-time quantum algorithm which can find the cut using O(n/є2) many copies of the state, which is optimal up to logarithmic factors. Our algorithm also generalizes to learn the entanglement structure of arbitrary product states. In the special case of Haar-random states, we further show that our algorithm requires circuits of only constant depth. To develop our algorithm, we introduce a state generalization of the hidden subgroup problem (StateHSP) which might be of independent interest, in which one is given a quantum state invariant under an unknown subgroup action, with the goal of learning the hidden symmetry subgroup. We show how the hidden cut problem can be formulated as a StateHSP with a carefully chosen Abelian group action. We then prove that Fourier sampling on the hidden cut state produces similar outcomes as a variant of the well-known Simon’s problem, allowing us to find the hidden cut efficiently. Therefore, our algorithm can be interpreted as an extension of Simon’s algorithm to entanglement testing. We discuss possible applications of StateHSP and hidden cut problems to cryptography and pseudorandomness.
In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use O(log(∣G∣)/ε) copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using O(log(∣G/H∣)/ε) forward and inverse queries, and prove a matching Ω(log(∣G/H∣)/ε) lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, Θ(log(∣G/H∣)/ε) copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in ε genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.
Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are, in fact, special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks and provide protocols that are easier to implement on near-term quantum devices.
How can we use a quantum computer to detect the entanglement structure of a quantum state? Bouland et al. (2024) recently provided an algorithm that, given multiple input copies of the state, finds the "hidden cuts"-partitions into fully unentangled qubit registers. Their solution is based on turning cuts into a symmetry which can be detected with a Shor-type quantum algorithm for hidden subgroup problems, the hidden cut algorithm. In this paper we derive heuristics that can find "approximate symmetries", or weakly entangled qubit registers, to unlock this powerful idea for a much broader range of problems. Our core contribution is a rigorous link between the output distribution of the hidden cut algorithm and the reward function that measures the quality of a cut. This implies that reducing the number of state copies in the original hidden cut algorithm leads to measurement samples from which patterns of weak entanglement can be extracted. We believe that these insights are an important step in making quantum algorithms for hidden subgroup problems useful for applications beyond cryptography.
Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of \emph{quantum} objects. Existing work in this area centres on the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We develop efficient quantum algorithms for non-abelian StateHSP when the hidden subgroup is normal and the ambient group belongs to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting.
StateHSP learns \emph{Bose} symmetries, under which a state must be invariant exactly under the action of the symmetry group. In quantum mechanics, however, physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a natural notion of \emph{Anyonic} state symmetry learning, based on invariance up to global phase. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction rests on a new correspondence between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups of mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and give efficient algorithms for them by reduction to state symmetry learning. Together, these results broaden the scope of state symmetry learning as a common algorithmic primitive for learning quantum symmetries.
Daniel Simon's 1994 discovery of an efficient quantum algorithm for finding “hidden shifts” of Z 2 n provided the first algebraic problem for which quantum computers are exponentially faster than their classical counterparts. In this article, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G , this is the problem of recovering an involution m = ( m 1 ,…, m n ) ∈ G n from an oracle f with the property that f ( x ⋅ y ) = f ( x ) ⇔ y ∈ {1, m }. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G n , where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G n have a simple product structure, they share important representation--theoretic properties with the symmetric groups S n , where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called “standard method” requires highly entangled measurements on the tensor product of many coset states. In this article, we provide quantum algorithms with time complexity 2 O (√ n ) that recover hidden involutions m = ( m 1 ,… m n ) ∈ G n where, as in Simon's problem, each m i is either the identity or the conjugate of a known element m which satisfies κ( m ) = −κ(1) for some κ ∈ Ĝ . Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the “missing harmonic” approach of Moore and Russell. These are the first nontrivial HSP algorithms for group families that require highly entangled multiregister Fourier sampling.
Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively.
In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form G=A⋊φZpk, with A finite Abelian, p prime, k∈N and the action of Zpk is generated by the scalar automorphism a↦μa, for some μ∈ZExp(A)×, where Exp(A) is the exponent of A. Our algorithm is efficient when A has bounded generator rank and Exp(A)/p=polylog(∣G∣). This includes the case A=ZN for N∈N and k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=Zqr with q prime and r∈N studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).