Entanglement bounds on the performance of quantum computing architectures

There are many possible architectures of qubit connectivity that designers of future quantum computers will need to choose between. However, the process of evaluating a particular connectivity graph's performance as a quantum architecture can be difficult. In this paper, we show that a quantity known as the isoperimetric number establishes a lower bound on the time required to create highly entangled states. This metric we propose counts resources based on the use of two-qubit unitary operations, while allowing for arbitrarily fast measurements and classical feedback. We use this metric to evaluate the hierarchical architecture proposed by A. Bapat et al. [Phys. Rev. A 98, 062328 (2018)] and find it to be a promising alternative to the conventional grid architecture. We also show that the lower bound that this metric places on the creation time of highly entangled states can be saturated with a constructive protocol, up to a factor logarithmic in the number of qubits.

Figure 1

Illustration of how a model with ancilla mediator qubits can be abstracted into one in which only data qubits and edge weights are tracked. In panel (a), each module (blue dashed circle) contains one data qubit (red) and several ancilla mediator qubits (green) that form Bell pairs with other modules. In panel (b), the module as a whole is represented by blue circles, while the ancilla mediator qubits are now represented by edge weights. Only the states of the data qubits are tracked.

Figure 2

An illustration of how a rainbow state is defined on an arbitrary subgraph $F$. Here, gray lines represent the connectivity graph of allowed two-qubit interactions, while doubled black lines represent maximally entangled qubit pairs. Qubits without a doubled line are assumed to be in state $|0\rangle$.

Figure 3

An illustration of the fictitious nodes added to the isoperimetric set, $F$, and a set of equal size $K$ (encircled by purple dashed line), to create a flow network. The new fictitious nodes, $s$ and $t$, appear as green triangles connected to every node in $F$ and $K$, respectively; the original nodes and edges are pink (in $F$) and blue (in $\overline{F}$) circles. The edges have weight one. The flow, shown by arrows, transfers $\lceil \left|F\right|/{\tau }_{\mathrm{RB}}\rceil =2$ units of entanglement across the bipartition. Gray, dotted edges are not used by the flow.

Figure 4

Examples of a hierarchical product (left) and a weighted hierarchy (right).

Figure 5

Three classes of subgraph used in our proof. Circles represent vertices in $F$, squares are vertices in $\overline{F}$, and dashed lines are edges in $\partial F$. (a) A situation in which part of one copy of $H_2$ is in $F$. (b) A situation in which the division between $F$ and $\overline{F}$ lies entirely in $H_1$. (c) A situation in which all but one of the copies of $H_2$ are entirely contained in $F$.