Sequential quantum processes with group symmetries

Abstract

Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a $(G \times H)$-invariant quantum comb for compact groups $G$ and $H$ using the corresponding Clebsch-Gordan transforms, which naturally extends to the $G$-covariant quantum comb. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation $U \in \mathrm{SU}(d)$ to its inverse $U^\dagger$ or transpose $U^\top$. From numerics, we find a deterministic and exact unitary transposition protocol for $d=3$ with $7$ queries to $U$. This protocol improves upon the protocol shown in the previous work, which requires $13$ queries to $U$.

Figures (8)

• Figure 1: (a) Quantum comb with $n-1$ open slots, where $\Psi_1, \ldots, \Psi_{n}$ are quantum channels. (b) Quantum comb transforms quantum channels $\Phi_1, \ldots, \Phi_{n-1}$ into a quantum channel $\Phi_\mathrm{out}$ given in Eq. \ref{['eq:comb_output']}. (c) Quantum comb can be considered as a quantum channel $\Psi: \bigotimes_{i=1}^{n} \mathop{\mathrm{End}}\nolimits(\mathcal{I}_i) \to \bigotimes_{i=1}^{n} \mathop{\mathrm{End}}\nolimits(\mathcal{O}_i)$ with non-signaling conditions $\mathcal{I}_i \not\to \mathcal{O}_j$ for $j<i$.
• Figure 2: (a) The $\mathop{\mathrm{U}}\nolimits(d)$-invariant channel can be implemented using the quantum Schur transforms and a quantum channel $\widetilde{\Psi}$ on the multiplicity space and the preparation of the maximally mixed state $\pi_\mu$. The ground symbol in the circuit represents the partial trace. This implementation cannot be streamed due to the presence of $\pi_\mu$. (b) The $\mathop{\mathrm{U}}\nolimits(d)$-invariant channel can also be implemented using the quantum Schur transform, the dual Schur transform and a quantum channel $\widetilde{\Psi}$ on the multiplicity space. This implementation can be streamed as shown in the SM supple.
• Figure 3: (a) Implementation of the quantum comb with the $(G\times H)$-invariance \ref{['eq:gh_symmetry']}, which is composed of isometries $W_i$ given in (b). The ground symbol in the circuit represents the partial trace. (b) The isometry $W_i$ for the $(G\times H)$-invariant quantum comb is given by the CG transforms and an arbitrary isometry operator $V_i^{\lambda_i \mu_{i-1}}$ shown in Eq. \ref{['eq:V_i']}.
• Figure 4: Comparison of the number of variables in the parametrized quantum comb for unitary transposition with $\mathop{\mathrm{U}}\nolimits(d)\times \mathop{\mathrm{U}}\nolimits(d)$ symmetry and that in the naive approach mo2025parameterized to reproduce the same circuit. The $x$-axis represents the query number $n$ and the $y$-axis represents the number of variables in log scale. The blue and orange lines represent the number of variables in the parametrized quantum comb with $\mathop{\mathrm{U}}\nolimits(d)\times \mathop{\mathrm{U}}\nolimits(d)$ symmetry and the naive approach, respectively.
• Figure 5: Example of a Bratteli diagram $\mathscr{B}_L$ for $\mathcal{A}^3_{n,1}$ adapted to $\mathcal{A}^d_{0,0} \hookrightarrow \mathcal{A}^d_{1,0} \hookrightarrow \mathcal{A}^d_{2,0} \hookrightarrow \mathcal{A}^d_{3,0} \hookrightarrow \mathcal{A}^d_{3,1}$. Matrix units $E^{\lambda}_{T,T'}$ from \ref{['6:def:Omega', 'eq:app4_ansatz_C']} are adapted to this chain.

Theorems & Definitions (15)

• Theorem 1
• proof : Proof sketch
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof