Random dilation superchannel

TL;DR

This work introduces an efficient quantum circuit that transforms n parallel queries of an unknown quantum channel into n parallel queries of a randomly chosen dilation isometry, generalizing the random purification concept to channels. The construction relies on Schur-Weyl duality, the quantum Schur transform, and the QFT over the symmetric group to realize a random purification, which is then extended to random dilation via a Choi-state approach and postselected teleportation. The main results include a poly(n, log d_I, log d_O) circuit for the random dilation superchannel under Kraus rank r = d_I/d_O, plus applications to storage-and-retrieval with exponential program-cost reduction, quantum channel superreplication (Θ(n^α) for α<2), probabilistic Petz recovery, and extensions to quantum superchannels. An alternative Kronecker-transform-based construction is also provided, offering potential pathways to sequential-query implementations, though the sequential case remains open for deeper exploration.

Abstract

We present a quantum circuit that implements the random dilation superchannel, transforming parallel queries of an unknown quantum channel into parallel queries of a randomly chosen dilation isometry of the input channel. This is a natural generalization of a random purification channel, that transforms copies of an unknown mixed state to copies of a randomly chosen purification state. Our construction is based on the quantum Schur transform and the quantum Fourier transform over the symmetric group. By using the efficient construction of these quantum transforms, we can implement the random dilation superchannel with the circuit complexity $O(\mathrm{poly}(n, \log d_I, \log d_O))$, where $n$ is the number of queries and $d_I$ and $d_O$ are the input and output dimensions of the input channel, respectively. As an application, we show an efficient storage-and-retrieval of an unknown quantum channel, which improves the program cost exponentially in the retrieval error $\varepsilon$. For the case where the Kraus rank $r$ is the least possible (i.e., $r = d_I/d_O$), we show quantum circuits transforming $n$ parallel queries of an unknown quantum channel $Λ$ to $Θ(n^α)$ parallel queries of $Λ$ for any $α<2$ approximately, and its Petz recovery map for the reference state given by the maximally mixed state probabilistically and exactly. We also show that our results can be further extended to the case of quantum superchannels.

Figures (6)

• Figure 1: Quantum circuit implementing the random purification channel $\Phi$ that transforms $n$ parallel queries of an unknown quantum state $\rho \in \mathcal{L}(\mathbb{C}^d)$ into $n$ parallel queries of a randomly chosen purification of $\rho$. See Eqs. \ref{['eq:schur_weyl_duality']}--\ref{['eq:def_pi_u']} for the definitions of the quantum Schur transform $U_\mathrm{Sch}$, the controlled permutation unitary $\mathrm{ctrl}-\pi$, the quantum Fourier transform over the symmetric group $\mathrm{QFT}_{\mathfrak{S}_n}$, the quantum state $\ket{+_{\mathfrak{S}_n}}$, and the maximally mixed state $\pi_{\mathcal{U}_{\lambda}}$.
• Figure 2: (a) Quantum circuit implementing the random dilation superchannel $\mathcal{C}$ that transforms $n$ parallel queries of an unknown quantum channel $\Lambda:\mathcal{L}(\mathcal{I}) \to \mathcal{L}(\mathcal{O})$ into $n$ parallel queries of a randomly chosen dilation isometry of $\Lambda$. (b) A proof idea of implementing the random dilation superchannel $\Xi$ using the closed timelike curve based on postselected teleportation (P-CTC) lloyd2011quantum. The symbol "$<$" represents the unnormalized maximally entangled state $\vert {\mathds{1}} \rangle\!\rangle$, and the symbol "$>$" represents the P-CTC. By replacing the controlled permutation channel $\mathrm{ctrl}-\pi$ sandwiched by $\vert {\mathds{1}} \rangle\!\rangle$ and the P-CTC with the transposed controlled permutation channel $\mathrm{ctrl}-\pi^{\mathsf{T}}$, we obtain the quantum circuit in (a).
• Figure 3: Quantum circuits implementing transformation of an unknown quantum channel $\Lambda:\mathcal{L}(\mathcal{I}) \to \mathcal{L}(\mathcal{O})$ with Kraus rank at most $r = d_I/d_O$. (a) Quantum superchannel that transforms $n$ parallel queries of $\Lambda$ into $N$ approximate parallel queries of $\Lambda$ for $N = \Theta(n^{\alpha})$ with any $\alpha<2$. (b) Probabilistic quantum superchannel that transforms $n = O(d_I^3/\eta)$ parallel queries of $\Lambda$ into the Petz recovery map $\mathcal{P}^\Lambda$ of $\Lambda$ for the reference state given by the maximally mixed state with the success probability at least $1-\eta$.
• Figure 4: Quantum circuit implementing another construction of the random dilation superchannel $\mathcal{C}$ that transforms $n$ parallel queries of an unknown quantum channel $\Lambda:\mathcal{L}(\mathcal{I}) \to \mathcal{L}(\mathcal{O})$ into $n$ parallel queries of a randomly chosen dilation isometry of $\Lambda$. The quantum gate $\mathrm{CG}_{\mathfrak{S}_n}$ represents the Kronecker transform with irreps specified by the controlled qubits. The control wire from the state $\ket{i_\mu}$ represents the irrep $\mu$. The quantum state $\ket{\phi^+_{\mathcal{S}_\mu}}$ is the maximally entangled state $\ket{\phi^+_{\mathcal{S}_\mu}} \coloneqq {1\over \sqrt{m_\nu}}\sum_{i} \ket{i}\otimes \ket{i}$ using the Young--Yamanouchi basis $\{\ket{i}\}_{i=1}^{m_\mu}$ of $\mathcal{S}_\mu$.
• Figure 5: Quantum circuit implementing the random dilation supersuperchannel $\Sigma$ that transforms $n$ parallel queries of an unknown quantum superchannel $\mathcal{C}$ into $n$ parallel queries of a randomly chosen dilation superchannel of $\mathcal{C}$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• proof : Proof of Thm. \ref{['thm:random-purification-channel']}
• proof : Proof of Thm. \ref{['thm:channel_superreplication']}
• proof : Proof of Thm. \ref{['thm:channel_petz_recovery']}