On the Fundamental Resource for Exponential Advantage in Quantum Channel Learning

TL;DR

Investigating the fundamental resources behind quantum advantage in channel learning reveals that while extensive entanglement is unnecessary, the number of ancilla qubits is crucial, and the dimension of the quantum memory is a crucial resource.

Abstract

Quantum resources enable us to achieve an exponential advantage in learning the properties of unknown physical systems by employing quantum memory. While entanglement with quantum memory is recognized as a necessary qualitative resource, its quantitative role remains less understood. In this work, we distinguish between two fundamental resources provided by quantum memory -- entanglement and ancilla qubits -- and analyze their separate contributions to the sampling complexity of quantum learning. Focusing on the task of Pauli channel learning, a prototypical example of quantum channel learning, remarkably, we prove that vanishingly small entanglement in the input state already suffices to accomplish the learning task with only a polynomial number of channel queries in the number of qubits. In contrast, we show that without a sufficient number of ancilla qubits, even learning partial information about the channel demands an exponentially large sample complexity. Thus, our findings reveal that while a large amount of entanglement is not necessary, the dimension of the quantum memory is a crucial resource. Hence, by identifying how the two resources contribute differently, our work offers deeper insight into the nature of the quantum learning advantage.

Figures (4)

• Figure 1: Schematic illustration of learning a quantum channel. (a) Learning a quantum channel acting on an $n$-qubit system without quantum memory. (b) Quantum channel learning with the assistance of quantum memory. The availability of quantum memory allows the use of $k$ ancilla qubits as a resource. Another resource, the entanglement entropy between the ancilla and the system, is denoted by $S_{\text{a}|\text{s}}$. The channel acts only on the system, while the ancilla is stored in the quantum memory. Joint measurements, such as Bell measurements, are also permitted.
• Figure 2: Illustration of the regime in which exponential sample complexity arises to accomplish the ($\varepsilon, \delta, w$)-Pauli channel learning task. The regime is denoted as a function of the maximum weight $w$ and the number of ancilla qubits $k$. In this figure, we focus on the case $k$ and $w$ scale proportionally with $n$. As stated in Theorem \ref{['theorem 2']}, the boundary of this regime is linear for $w \leq n/2$, and becomes concave for $w > n/2$. According to Eq. \ref{['lower bound on CN(P(w))']}, within our stabilizer-covering scheme, polynomial sample complexity is achievable only when $k=n$. This case is indicated by the black dashed line.
• Figure 3: Illustration of the set $\mathsf{G}^{(n)}(\mathbb{G})$. Each number in the box labels a qubit. $G_{j}$ denotes the $j$-th element of $\mathbb{G}$, and $\mathbf{g}^{(j)}$ is an element of $\mathsf{G}^{(n)}(\mathbb{G})$ that applies $G_{j}$ to qubit $j$ and the identity elsewhere.
• Figure 4: Illustration of $\mathbf{g}$, $\mathsf{G}^{ (\mathrm{A}) }(\mathbf{g}, \mathcal{A})$, and $\mathsf{G}^{ (\mathrm{B}) }(\mathbf{g}, \mathcal{B})$. Each boxed number indicates the corresponding qubit index. Although we draw $\mathcal{A} = \{\mathcal{A}_{j}\}_{j=1}^{2(n-w)}$ and $\mathcal{B} = \{\mathcal{B}_{j}\}_{j=1}^{2w-n}$ as contiguous subsets of qubits for simplicity, all of two subsets are considered. Each element of $\mathsf{G}^{ (\mathrm{A}) }(\mathbf{g}, \mathcal{A})$ is denoted by $\mathbf{a}^{ (j) }$, whose $\mathcal{A}_{j}$-th component is $g_{ \mathcal{A}_{j} }$, and all other components are $I$. We denote each element of $\mathsf{G}^{ (\mathrm{B}) }(\mathbf{g}, \mathcal{B})$ by $\mathbf{b}^{ (j) }$, where $b_{\mathcal{B}_{j}}^{ (j) } = \mathcal{P}(g_{ \mathcal{B}_{j} })$, $b_{\mathcal{B}_{j+1}}^{ (j) } = \mathcal{P}(g_{ \mathcal{B}_{j+1} })$, and all other components are $I$. Here, $\mathcal{P}(X) = Y$, $\mathcal{P}(Y) = Z$, and $\mathcal{P}(Z) = X$.

Theorems & Definitions (4)

• Definition 1: ($\varepsilon, \delta, w$)-Pauli channel learning task
• Theorem 1
• Theorem 2
• Theorem 3