Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations

TL;DR

Quantum combs provide a framework to transform quantum processes, but solving for the optimal comb via SDP scales exponentially with the number of slots as ${\cal O}(d^{4m})$. To address this, the authors propose PQComb, which uses parameterized quantum circuits (PQC) to replace each comb tooth and trains the overall circuit through loss functions ${\cal L}_p$ or ${\cal L}_c$, enabling scalable learning of target transformations. Applied to unknown qubit unitaries, PQComb yields a 3-ancilla, 4-call deterministic inversion achieving $U_{\text{in}}^{-1}$, with further refinement toward an exact $(m,n_a)$ protocol and extensions to qutrit inverse/transpose and channel discrimination. Hardware-style simulations on IBM-Q demonstrate improved resource efficiency and robustness against realistic noise, underscoring practical viability on near-term devices. Overall, PQComb offers a flexible, data-driven pathway to learn and deploy quantum transformation protocols beyond SDP, with potential impact on quantum machine learning and higher-dimensional quantum information processing.

Abstract

Quantum combs play a vital role in characterizing and transforming quantum processes, with wide-ranging applications in quantum information processing. However, obtaining the explicit quantum circuit for the desired quantum comb remains a challenging problem. We propose PQComb, a novel framework that employs parameterized quantum circuits (PQCs) or quantum neural networks to harness the full potential of quantum combs for diverse quantum process transformation tasks. This method is well-suited for near-term quantum devices and can be applied to various tasks in quantum machine learning. As a notable application, we present two streamlined protocols for the time-reversal simulation of unknown qubit unitary evolutions, reducing the ancilla qubit overhead from six to three compared to the previous best-known method. We also extend PQComb to solve the problems of qutrit unitary transformation and channel discrimination. Furthermore, we demonstrate the hardware efficiency and robustness of our qubit unitary inversion protocol under realistic noise simulations of IBM-Q superconducting quantum hardware, yielding a significant improvement in average similarity over the previous protocol under practical regimes. PQComb's versatility and potential for broader applications in quantum machine learning pave the way for more efficient and practical solutions to complex quantum tasks.

Figures (9)

• Figure 1: The overview of training formalism for the parameterized quantum comb framework. Within this scheme, the channel for the $k$-th tooth ${\cal V}_k(\theta_k)$ is now parameterized by $\theta_k$ that remains tunable to adjustments during the optimization phase, and $\bm \theta = (\theta_0, \ldots, \theta_m)$ is denoted as the vector of all parameters in this PQComb. (a) describes how the PQComb trains the protocol using the process-based loss function ${\cal L}_p$, which is computed by the average dissimilarity between the sampled output process $\widehat{{\cal N}}_{j, \textrm{out}}(\bm \theta)$ and the expected process ${\cal N}_{j, \textrm{out}}$. (b) describes how the PQComb trains the protocol using the comb-based loss function ${\cal L}_c$, which optimizes the Choi operator of the circuit $C_ \mathbf{V} (\bm \theta)$ using the performance operator $\Omega$. Here each pair of two dots connected by a line represents the unnormalized maximally entangled state.
• Figure 2: The proposed unitary inversion protocol for arbitrary single-qubit unitary $U_\textrm{in}$. One can use 3 ancilla qubits and 4 queries of $U_\textrm{in}$ to realize qubit-unitary inversion. Note that the output state of the first ancilla qubit will be a zero state without post-selection. The implementations of ${\cal Q}_{ U_\textrm{in} }$ and $G$ are deferred to Supplementary Note 2 Note2.
• Figure 3: Simulation of our two protocols and the previous protocol in Ref. Yoshida2023 under the noise settings of five real quantum devices. Here we refer to the protocols in Theorem \ref{['thm:4 3']} and Corollary \ref{['coro:5 3']} as the "4-call protocol" and "5-call protocol", respectively.
• Figure S1: The schematic representation of a sequential quantum comb $\mathfrak{C}$.
• Figure S2: Circuit implementation of $\mathfrak{C}_\textrm{SWAP}$, where $\textrm{SWAP}_d$ is the generalized $d^2$-dimensional SWAP gate

Theorems & Definitions (4)

• Theorem 1: 3-ancilla 4-call Protocol
• Corollary 2: 3-ancilla 5-call Protocol
• Theorem S1
• Corollary S2