Lightcone Bounds for Quantum Circuit Mapping via Uncomplexity

TL;DR

This work reconstitute quantum circuit mapping using tools from quantum information theory, showing that a lower bound emerges for a circuit executed on hardware, and develops an initial placement algorithm based on graph similarity search, aiding in optimally placing circuit qubits onto a device.

Abstract

Efficiently mapping quantum circuits onto hardware is an integral part of the quantum compilation process, wherein a circuit is modified in accordance with the stringent architectural demands of a quantum processor. Many techniques exist for solving the quantum circuit mapping problem, in addition to several theoretical perspectives that relate quantum circuit mapping to problems in classical computer science. This work considers a novel perspective on quantum circuit mapping, in which the routing process of a simplified circuit is viewed as a composition of quantum operations acting on density matrices representing the quantum circuit and processor. Drawing on insight from recent advances in quantum circuit complexity and information geometry, we show that a minimal SWAP-gate count for executing a quantum circuit on a device emerges via the minimization of the distance between quantum states using the quantum Jensen-Shannon divergence, which we dub the lightcone bound. Additionally, we develop a novel initial placement algorithm based on a graph similarity search that selects the partition nearest to a graph isomorphism between interaction and coupling graphs. From these two ingredients, we construct an algorithm for calculating the lightcone bound, which is directly compared alongside the IBM Qiskit compiler for over $600$ realistic benchmark experiments, as well as against a brute-force method for smaller benchmarks. In our simulations, we unambiguously find that neither the brute-force method nor the Qiskit compiler surpasses our bound, signaling utility for estimating minimal overhead when realizing quantum algorithms on constrained quantum hardware. This work also constitutes the first use of quantum circuit uncomplexity to practically-relevant quantum computing. We anticipate that this method may have diverse applicability outside of the scope of quantum information science.

Figures (10)

• Figure 1: An example of the QCMP as a sequence of steps needed to assign qubits from an algorithm to a quantum device. The two-qubit gates in the two circuit diagrams are used to represent general two-qubit unitary operations (with the exception of SWAP gates); here, we do not consider single-qubit gates, and the two-qubit interactions shown in (a) and (d) are taken to be general two-qubit unitary operations. (a) The quantum circuit is transformed into an interaction graph, as shown in (b). Next, it is compared with the connectivity properties of the coupling graph (c). As no graph isomorphism (i.e., no exact matching between the vertices of the IG and CG which upholds all of the edge relations of both) exists between the IG and CG, one can compensate for the lack of connectivity by introducing SWAP operations to the circuit in order to realize the circuit. (d) These operations degrade the fidelity of the final output state.
• Figure 2: The fluctuation of $\mathcal{S}_{G}(\bm{\rho_{i}})$ as a function of $\beta$ for $4$-node graphs. We have utilized different marker types in order to distinguish curves with very similar VNE.
• Figure 3: Steps taken for the qubit-assignment algorithm described in \ref{['section:qubit_assignment']}: (a) The task at hand is to find the best-fit initial placement for the qubits in the $4$-qubit IG (shown with blue numbering) for the $7$-qubit CG (shown with red numbering); in our case, the the CG corresponds to the connectivity of the IBM Casablanca quantum device. (b) Shows the distinct subgraphs found from \ref{['step:identify']} in \ref{['section:qubit_assignment']}. After verifying that no direct subgraph isomorphism between IG and one of these graphs exists, a similarity search is employed. (c) The subgraph of the CG with the lowest GED relative to an IG is retrieved. The resulting initial placement is shown in blue, with the final GED calculated to be $1$. The actual qubit assignment is shown in the form of several colored ordered pairs (blue numbering represents IG qubits, while red numbering represents CG qubits).
• Figure 4: Simulation results for various IG / CG benchmark pairs. The horizontal axis enumerates each benchmark circuit tested (sorted by the number of two-qubit gates), and the vertical axis describes the normalized number of SWAPs, due to very high maximal bounds (the SWAP-bound values of each benchmark are divided by their sum). The results are color-coded as follows: the SWAP uncomplexity of \ref{['section:thermo_path_length_uncomplexity_derivation']} (green); the Qiskit compiler with default options Sabre router and circuit optimization level $1$Qiskit (orange); and the maximal maximal bound calculated as in \ref{['section:maxboundcalc']} (blue). In every IG/CG pair, the bound calculated captures the SWAP uncomplexity that is either approachable or unattainable by the Qiskit compiler, thus empirically demonstrating our formulation.
• Figure 5: Subset of simulation results from \ref{['figure:swapbounds']} where only one CG is shown (Google Bristlecone). Benchmark circuits are sorted by a) number of two-qubit gates; and b) IG complexity (number of nodes and edges of IGs). The respective benchmarks with their respective nodes-edges count are detailed in \ref{['tab:benchmarks']}.