QuarterMap: Quantum Circuit Mapper

• QuarterMap is an open-source tool for quantum circuit mapping that optimally assigns logical qubits to physical hardware while minimizing added SWAP gates.
• It employs dual strategies—an exact mapper using MaxSAT with Z3 for optimal solutions on small circuits and a heuristic A* search for scalable mapping on larger instances.
• Integrated in the Munich Quantum Toolkit, QuarterMap bridges algorithm design with hardware constraints, supporting platforms like IBM devices and Python-based quantum stacks.

Searching arXiv for "QuarterMap" and closely related terms to ground the article in current paper records. Searching arXiv for "MQT QMAP", "Quarter sampling", and "quarter plane conformal map". QuarterMap, in the toolkit context, refers to MQT QMAP, an open-source quantum circuit mapping tool in the Munich Quantum Toolkit (MQT). Its purpose is to transform a circuit formulated on logical/virtual qubits into a circuit executable on a target device with physical qubits and a restricted connectivity graph, while keeping mapping overhead low. The tool is positioned as an efficient, automated, and accessible solution to the quantum circuit mapping problem and combines an exact mapper for optimal solutions on small instances with a heuristic mapper for larger instances (Wille et al., 2023).

1. Definition and problem domain

Quantum algorithms are commonly specified as gate sequences acting on abstract qubits, often under the implicit assumption that any pair of qubits can interact. Actual quantum hardware imposes two independent constraints. First, the circuit must be expressed in the device’s native gate set. Second, two-qubit gates must respect the hardware topology / connectivity constraint, meaning that such gates can only be applied to qubit pairs connected in the device’s coupling map. QuarterMap addresses the second problem directly by assigning logical qubits to physical qubits and inserting routing operations so that all required interactions become executable (Wille et al., 2023).

The problem is operationally centered on qubit allocation / initial layout and routing. Allocation determines where each logical qubit starts on the hardware. Routing then moves qubits indirectly, typically by inserting SWAP gates, until interacting qubits become adjacent or otherwise compatible with the architecture. Because present-day devices are highly error-sensitive, every additional operation affects execution quality. For that reason, QuarterMap treats mapping quality in terms of costs such as (two-qubit) gate count, circuit depth, and expected fidelity, with particular emphasis on the number of added SWAPs (Wille et al., 2023).

2. Optimization objective and hardware model

The quantum circuit mapping task is described as NP-hard. In practical terms, the optimization target is to satisfy connectivity constraints while minimizing added overhead, especially the routing overhead introduced during compilation. In the formulation highlighted by QuarterMap, this is closely approximated by

$$\text{minimize mapping overhead} \quad \approx \quad \text{minimize added SWAP gates}.$$

The exact mapper is explicitly characterized as producing gate-optimal solutions, i.e., solutions with the minimal number of SWAP gates (Wille et al., 2023).

The hardware model used by QuarterMap focuses mainly on superconducting qubit architectures, where sparse connectivity is a dominant systems constraint. Connectivity is represented by a coupling graph / coupling map, whose edges indicate which qubit pairs can participate in a two-qubit gate. The graph may be represented as directed, but the paper notes that it is often effectively undirected in practice. IBM devices are used as reference architectures, including ibm_washington, a 127-qubit device with sparse connectivity, and ibm_oslo, a 7-qubit architecture used in an example walkthrough. Sparse coupling is the core reason routing becomes necessary at all (Wille et al., 2023).

This hardware framing makes QuarterMap part of the compilation layer that mediates between algorithm-level circuit design and architecture-level execution constraints. A plausible implication is that its value lies less in altering circuit semantics than in controlling the compilation penalty induced by realistic machine topologies.

3. Mapping strategies

QuarterMap provides two principal algorithmic modes. One targets optimality on small instances; the other targets scalability on larger workloads. The distinction is fundamental to the tool’s design (Wille et al., 2023).

Strategy	Core method	Scope
Exact mapping	MaxSAT encoding solved with Z3	Minimal number of SWAP gates; typically scalable only up to roughly 8 qubits and around 1000 gates
Heuristic mapping	A* search	Circuits with hundreds of qubits and hundreds of thousands of gates

The exact mapper encodes the mapping task as a MaxSAT instance and solves it with the Z3 SMT solver. Its defining property is optimality with respect to added SWAP count. This comes at substantial computational cost, so the method is suitable only for relatively small circuits. The paper identifies two search-space reduction ideas used internally: restricting permutations before each layer, so that only enough permutations are considered to make interacting qubits adjacent, and restricting subarchitectures, so that search is confined to viable hardware subgraphs rather than the full architecture (Wille et al., 2023).

The heuristic mapper is based on A* search and is intended for cases where exact optimization is infeasible. It decomposes mapping into initial layout and routing, and supports three initial layout strategies: identity layout, where logical qubit $q_i$ maps to physical qubit $Q_i$; static layout, which inspects the first circuit layer and places interacting qubits on connected hardware qubits; and dynamic layout, which assigns qubits greedily on demand during mapping. The paper emphasizes that better initial layouts can materially improve routing quality (Wille et al., 2023).

A notable current limitation is that the heuristic mapper assumes the circuit’s two-qubit gates are CNOTs. The authors identify this as a limitation and a natural extension point for alternative hardware-native gate sets such as CZ (Wille et al., 2023).

4. Software architecture and workflows

QuarterMap is designed to be usable both from a research-user perspective and from a developer perspective. At the implementation level, the core is written in C++, exposed through Python bindings, and relies on MQT QFR for quantum functionality representation. Development requires C++17, CMake 3.19+, and Z3 4.8.15+ for the exact mapper. The codebase also includes unit tests, linting and formatting tools, and CI workflows across major operating systems (Wille et al., 2023).

From the user side, the tool is intended to integrate naturally with Python-based quantum software stacks. Architectures can be specified either directly from a Qiskit backend or by constructing them from a coupling map. Input circuits can be provided using Qiskit or formats such as OpenQASM. Mapping is exposed as a one-line compile call, returning both the mapped circuit as a Qiskit QuantumCircuit and a result object containing statistics such as the number of added SWAPs and runtime. The documentation is listed at mqtqmap.readthedocs.io, and the software is publicly available as open source at https://github.com/cda-tum/qmap (Wille et al., 2023).

This dual orientation—research-grade algorithms with accessible packaging—explains the paper’s emphasis on QuarterMap as both an algorithmic contribution and a software-engineering artifact. It is meant to be installable, interoperable, and extensible, not merely a proof-of-concept mapper.

5. Example mapping results and trade-offs

The paper illustrates QuarterMap on a 5-qubit VQE-style circuit mapped to the 7-qubit ibm_oslo architecture. The example is used to compare naive compilation, exact mapping, and heuristic mapping under the same topology (Wille et al., 2023).

A naive mapping introduces 4 SWAP gates and 1 additional qubit. The exact mapper finds a solution with 0 SWAP gates and 0 additional qubits, but requires on the order of seconds. The heuristic mapper produces a solution with 2 SWAP gates and 1 additional CNOT gate, while running in less than a millisecond. The example therefore exhibits the principal design trade-off explicitly: exact = better quality, slower; heuristic = slightly worse quality, much faster (Wille et al., 2023).

The significance of this example is not limited to the specific circuit. It demonstrates that mapping quality can vary sharply even for modest instances, and that the cost function relevant to near-term hardware is not abstract asymptotic complexity alone. Extra SWAPs, extra CNOTs, and extra qubit usage translate directly into greater noise exposure and lower expected fidelity. QuarterMap therefore occupies a practical optimization layer where algorithmic solvability and physical executability meet.

The paper also notes unresolved structural issues in the exact approach. A brute-force treatment of subarchitectures remains expensive, subarchitecture isomorphism is not fully exploited, and considering more qubits may sometimes yield better optimal mappings. These observations show that even the “optimal” mode is optimal relative to a particular encoding and search strategy rather than a fully exhausted architectural meta-space (Wille et al., 2023).

6. Terminological scope and related usages

Within the supplied arXiv corpus, QuarterMap is most directly grounded as the toolkit-context name for MQT QMAP. The same corpus also shows that the label is not uniformly standardized across other research areas. In the paper on discrete and continuous polyharmonic functions in the quarter plane, there is no symbol or named object “QuarterMap”; instead, the structurally analogous object is a conformal map $w$ used to solve quarter-plane boundary value problems (Nessmann, 2022). In the paper on compander-aligned zeroth-order optimization, the authors likewise state that the paper does not mention the term, although it studies a quantizer-induced map

$Q=\phi^{-1}\circ U\circ \phi$

and a map-aware query rule (Shu et al., 11 May 2026).

This suggests that QuarterMap is best understood as a toolkit-specific designation rather than a cross-disciplinary technical term with a single established definition. In quantum compilation, it denotes a concrete software system for circuit mapping. In other areas, superficially similar expressions involving “quarter,” “map,” or mapping on quarter domains refer to unrelated analytic, geometric, or sensing constructions. Recognizing this distinction avoids a common misconception: QuarterMap is not, in the present usage, a generic quarter-plane conformal map, a quarter-sampling reconstruction method, or a quantizer-aware optimization map, even though those topics also involve mathematically meaningful mappings (Wille et al., 2023).

In that sense, QuarterMap belongs to the vocabulary of practical quantum compilation: a mapping tool that takes the abstract combinatorics of a circuit and reconciles them with the sparse connectivity and execution constraints of actual quantum hardware.