Solving approximate hidden subgroup problems: quantum heuristics to detect weak entanglement

Abstract

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.

Figures (11)

• Figure 1: Finding weakly entangled bipartitions of qubits means finding subsystems $\bm s$ whose reduced state has a high purity $P(\bm s)$. We show here that the output distribution of the hidden cut algorithm run with $2t$ copies bouland2025state is the Fourier transform of $P^t(\bm s)$. Using many input copies attenuates all purities smaller than $1$ and guarantees that postprocessing the measurement samples reveals $\bm s$ with $P(\bm s)=1$ (top left). However, it also suppresses information on "approximate cuts" or $P(\bm s)<1$ (bottom left). To build heuristics, we need to use a small number of input copies, which keeps information on secondary maxima, but still suppresses low values (bottom right). This example uses $t=300$ (left) and $t=5$ (right) for a product of two Haar random states, $\ket{\psi} = \ket{\psi_{01}}\ket{\psi_{23}}$, which we mixed in $0.1$ parts with a Haar random state to create a weak entanglement bipartition at $\bm s = 0011$, $\bar{\bm s}=1100$. The code to reproduce all figures can be found here: https://github.com/XanaduAI/approximate_hidden_subgroups.
• Figure 2: StateHSP circuit for group $G$.
• Figure 3: General $t$-copy hidden cut circuit.
• Figure 4: Non-separable state $\ket\psi=\ket{\psi_{0123}}$: Subsystem purities $P^t(\bm s)$ and hidden cut circuit output distribution $p_t(\bm x)$ for a Haar-random 4-qubit state, which are related by a Fourier transform. The output distribution of the hidden cut algorithm has support over bitstrings that are orthogonal to bitstrings for which $P(\bm s)=1$. Here, the support is over all bitstrings of even parity, which are only orthogonal to $\bm s = 0000$ and $\bm s = 1111$. Increasing $t$ in the algorithm by adding state copies accentuates these features: $p_t$ becomes a uniform distribution over the support of $p_1$, and $P^t$ only keeps support for subsystems of purity $1$.
• Figure 5: Separable state $\ket\psi=\ket{\psi_{01}}\ket{\psi_{23}}$: Subsystem purities $P^t(\bm s)$ and hidden cut circuit output distribution $p_t(\bm x)$ for a product of two Haar-random 2-qubit state. The output distribution of the hidden cut algorithm for this state has support on the bitstrings $0000$ and $1111$ as well as $0011$ and $1100$. The same bitstrings are orthogonal to these, and represent the subsystems with purity $1$, hence revealing the "hidden cut".

Theorems & Definitions (2)

• Lemma 1
• proof