Space-time tradeoff in networked virtual distillation

TL;DR

This paper investigates how virtual distillation (VD) can mitigate quantum errors in networked quantum systems by analyzing three edge-case implementations that span space-time tradeoffs: cyclic rotation (CR), qubit-efficient cyclic rotation (QECR), and brickwork (BW). It derives resource and depth characterizations, proposes architectures for distributed VD across ion-trap networks, and demonstrates via realistic noise simulations that VD can suppress errors even with noisy states, with BW offering the strongest performance under practical constraints. The work also discusses remote operations, connectivity, and fault-tolerant integration, showing that local gate fidelity principally limits performance while remote entanglement bottlenecks are comparatively less severe. Overall, VD remains a promising technique for near-term and early fault-tolerant quantum devices, especially when copies are prepared in parallel and modularity is leveraged. The insights provide a roadmap for deploying VD in distributed quantum networks and in conjunction with quantum error correction.

Abstract

In contrast to monolithic devices, modular, networked quantum architectures are based on interconnecting smaller quantum hardware nodes using quantum communication links, and offer a promising approach to scalability. Virtual distillation (VD) is a technique that can, under ideal conditions, suppress errors exponentially as the number of quantum state copies increases. However, additional gate operations required for VD introduce further errors, which may limit its practical effectiveness. In this work, we analyse three practical implementations of VD that correspond to edge cases that maximise space-time tradeoffs. Specifically, we consider an implementation that minimises the number of qubits but introduces significantly deeper quantum circuits, and contrast it with implementations that parallelise the preparation of copies using additional qubits, including a constant-depth implementation. We rigorously characterise their circuit depth and gate count requirements, and develop explicit architectures for implementing them in networked quantum systems -- while also detailing implementations in early fault-tolerant quantum architectures. We numerically compare the performance of the three implementations under realistic noise characteristics of networked ion trap systems and conclude the following. Firstly, VD effectively suppresses errors even for very noisy states. Secondly, the constant-depth implementation consistently outperforms the implementation that minimises the number of qubits. Finally, the approach is highly robust to errors in remote entangling operations, with noise in local gates being the main limiting factor to its performance.

Figures (13)

• Figure 1: (a) Quantum circuit implementing VD using $n$ copies of the noisy quantum state $\rho$ (dark blue) and a single ancilla qubit (light blue) initialised in $\ket{+}=(1/\sqrt{2})(\ket{0}+\ket{1})$. A derangement operation is applied to all $n$ copies (denoted by $D_n$), conditioned on the state of the ancilla qubit, and is followed by a set of controlled Pauli gates (denoted by $\sigma$) associated with each term in the Pauli basis expansion of $O$. A measurement of the ancilla qubit in the $X$-basis reveals the nonlinear functional Tr$[\sigma\rho^n]$, which is central to the scheme. (b) Conceptual illustration of applying VD with 4 copies of 6 data qubits across a 4-node quantum network, where each copy (dark blue) is placed in a distinct node. The ancilla qubit (light blue) is placed in one of the nodes, and each node has an additional network qubit (orange) used to establish remote entanglement between nodes. With Bell state generation, local operations, and classical communication enabled, VD can be applied across the quantum network.
• Figure 2: $n$-cycles applied as (a) cyclic rotation and (b) brickwork, shown for the example where $n = 4$.
• Figure 3: Three implementations of VD after replacing the final layer of C-SWAP gates with BSMs: (a) Cyclic rotation (CR), (b) qubit-efficient cyclic rotation (QECR), and (c) brickwork (BW). The components highlighted in orange are present (absent) when the number of copies is odd (even). In every case, the controlled derangement in \ref{['fig: vd_a']} is decomposed into C-SWAP gates and BSMs, where the latter consists of a CNOT gate followed by $X$- and $Z$-basis measurements. The wavy line across multiple ancilla qubits represents a GHZ state. The amount of quantum resources used in each implementation is summarised in \ref{['tab:tradeoff_table']}.
• Figure 4: Connectivities between nodes required to apply VD with 6 or 7 copies across a quantum network, where one copy is stored in each node at every moment, using (a) CR (b) QECR and (c) BW implementations. Each white circle represents a node containing a single copy, and those with a black solid circle also includes an ancilla qubit. Bell states can be prepared between each pair of nodes that are connected by a line. Components highlighted in orange are present (absent) when considering 7 (6) copies.
• Figure 5: Remote operations used to apply VD across a quantum network. (a) Teleportation of the state of qubit $q_1$ to a qubit $q_2$ in a different node, given a Bell pair shared between qubits $a_1$ and $q_2$. (b) BSM between qubits $q_1$ and $q_2$ in different nodes. A remote BSM outcome of both 1 corresponds to when both pair of qubits $(q_1,a_2)$ and $(a_1,q_2)$ produce odd parity outcomes. (c) Preparation of an $n$-qubit GHZ state in constant time -- example for $n=4$ where the qubits $q_1$, $q_2$, $q_3$, and $q_4$ are located in distinct nodes and ancilla qubits $a_2$ and $a_3$ are used. The $X$ gates are corrections applied based on the collective outcome of the $Z$-basis measurements. The scheme scales straightforwardly to $n>4$ qubit GHZ states.