Design Automation Tools and Software for Quantum Computing: Inside the Munich Quantum Toolkit
Description
This book provides an easy-to-read introduction to quantum computing and introduces readers to a set of software design tools that leverage the power of design automation for optimizing quantum circuits and applications. The authors describe methods and software for quantum computing that utilize explicitly core data structures (such as decision diagrams) or methods (such as dedicated reasoning engines), while focusing on three major design tasks in quantum software development, namely classical simulation of quantum circuits, compilation of quantum circuits, and verification of quantum circuits. All tools and software solutions discussed in this book are made available as open-source packages and constitute the backbone of the Munich Quantum Toolkit.
Lukas Burgholzer works as a research scientist at the Technical University of Munich in the Chair for Design Automation of Robert Wille and as CTO of the Munich Quantum Software Company (MQSC). He received his PhD from JKU Linz, Austria, in 2023 working as part of the Institute for Integrated Circuits as well as the LIT Secure and Correct Systems Lab. His research focuses on design automation tools and software for quantum computing. In these areas, he has published more than 60 papers in international conferences and journals. He is the chief developer of the Munich Quantum Toolkit (MQT) as well as one of the technical leads of the Munich Quantum Software Stack (MQSS) project, which is developed as part of the Munich Quantum Valley (MQV) initiative. For his research, he was awarded the EDAA Outstanding Dissertation Award, the Heinz Zemanek Prize, and more.
Robert Wille is a Full and Distinguished Professor at the Technical University of Munich, CEO of the Munich Quantum Software Company (MQSC), and Scientific Director at the Software Competence Center Hagenberg. He received his Diploma and Dr.-Ing. degrees in Computer Science from the University of Bremen, Germany, in 2006 and 2009, respectively. His academic journey has included positions at the University of Bremen, the German Research Center for Artificial Intelligence (DFKI), the University of Applied Sciences Bremen, the University of Potsdam, and the Technical University Dresden. From 2015 to 2022, he worked as a Full Professor at the Johannes Kepler University Linz before moving to Munich.
His research focuses on the design of circuits and systems for both conventional and emerging technologies. For over 15 years, he has been actively advancing the field of quantum computing—establishing foundational software and design automation concepts. His contributions have earned numerous accolades, including Best Paper Awards, the DAC Under-40 Innovator Award, a Google Research Award, an ERC Consolidator Grant, and more. He collaborates with leading academic and industrial partners and plays a key role in initiatives such as the Munich Quantum Valley. He has published over 400 papers and serves on editorial boards and numerous boards of major journals and conferences. Through his roles in companies, he is also deeply engaged in technology transfer from research to practice.
See this answer to the Quantum Computing StackExchange question "How are quantum circuits run on real quantum computers?"
Also: Geremia, “Answer to ‘Are there classical tools for simulating ZX-calculus diagrams as if they were quantum circuits?’,” Quantum Computing Stack Exchange , July 3, 2026.
1. Introduction
05/18/261. Introduction : 26
the quantum state and the operations are commonly represented by vectors and matrices that have an exponential size with respect to the number of qubits
05/18/261. Introduction : 31
dynamic quantum circuits—circuits that use classical feedback during the execution of the circuit to influence the computation
3.3 ZX-Calculus
07/02/263.3 ZX-Calculus : 74
Any quantum circuit can be interpreted as a ZX-diagram.
07/02/263.3 ZX-Calculus : 75
The real power of ZX-diagrams becomes evident when adding rewrite rules to the language. The axioms of the scalar-free ZX-calculus are given in Fig. 3.1.
Fig. 3.1Axioms of the scalar-free ZX-calculus
07/02/263.3 ZX-Calculus : 76
rule, which is sometimes listed explicitly among the axioms for the ZX-calculus, needs to be proven first:
(3.31)
07/02/263.3 ZX-Calculus : 76
a dedicated ZX software package has been developed in C++ and is publicly available as part of the MQT Core library
07/02/263.3 ZX-Calculus : 77
Quantinuum’s TKET (https://github.com/CQCL/tket
alt. MQT Core’s ZX functionality
3.4 SAT Solving
05/30/263.4 SAT Solving : 77
any NP problem can be reduced to SAT in polynomial time
05/21/263.4 SAT Solving : 82
R. Wille, L. Burgholzer, M. Artner, Visualizing decision diagrams for quantum computing, in Design, Automation and Test in Europe (2021)
4. Overview of Part II
05/30/264. Overview of Part II : 88
the
05/30/264. Overview of Part II : 88
the importance of the order in which individual decision diagram computations are performed is studied
5.1.1 Limitations of Decision Diagram-Based Simulation
06/10/265.1.1 Limitations of Decision Diagram-Based Simulation : 95
Google in their quantum supremacy experiment [19, 20]).
5.3 Summary of Results
06/10/265.3 Summary of Results : 115
19.S. Boixo, S.V. Isakov, V.N. Smelyanskiy, et al., Characterizing quantum supremacy in near-term devices. Nat. Phys. 14(6), 595–600 (2018). arXiv: 1608.00263
20.S. Boixo, C. Neill, The question of quantum supremacy. Google AI Blog. (2018). https://github.com/sboixo/GRCS
6. Simulation Paths for Quantum Circuit Simulation with Decision Diagrams
06/16/266. Simulation Paths for Quantum Circuit Simulation with Decision Diagrams : 116
efficiency is heavily dependent on the order in which the computations are performed
reversibility
6.1 Motivation and Related Work
06/16/266.1 Motivation and Related Work : 118
the order in which the respective multiplications involved in classically simulating a quantum circuit are conducted influences the complexity of simulation methods based on decision diagrams
6.1.2 Related Work
06/16/266.1.2 Related Work : 124
Decision diagrams are not the only data structure for simulating quantum circuits that suffers from the exponential difference in best- and worst-case complexity.
6.2.2 Utilizing Research on Tensor Network Contraction
06/16/266.2.2 Utilizing Research on Tensor Network Contraction : 133
It allows one to determine contraction plans for large tensor networks using various graph-based methods and is publicly available at github.com/jcmgray/cotengra.
06/16/266.2.2 Utilizing Research on Tensor Network Contraction : 133
Example 6.5Feeding the tensor network shown in Example 6.4 into CoTenGra results in the following simulation path:
[(0,1),(2,8),(3,9),(4,10),(5,11),(6,12),(7,13)]
(6.13)The resulting contraction tree can be visualized as
(6.14)To this end, the color of each node represents the number of floating point operations required for a particular contraction, while the color of each edge represents the size of the respective tensors—the darker the color, the more complex the contraction or the larger the tensor.
Does CoTenGra use Dijstra?
7.2 Desired Simulation Output
06/17/267.2 Desired Simulation Output : 150
Decision diagrams are more suited towards full state vector simulation
7.3 Determining the Order of Computation
06/17/267.3 Determining the Order of Computation : 153
no clear runtime estimation can be derived for classical simulation based on decision diagrams.
9.1 The Quantum Circuit Compilation Flow
06/17/269.1 The Quantum Circuit Compilation Flow : 169
OQC
= Oxford Quantum Circuits
11.5 Summary of Results
06/17/2611.5 Summary of Results : 226
18.N.C. Carter, Visual Group Theory (Mathematical Association of America, 2009)
13. Overview of Part IV
06/17/2613. Overview of Part IV : 232
Quantum Verification, i.e., checking whether a device indeed performs quantum logic (see [8]) and, therefore, indeed utilizes quantum advantage,
13.2 Equivalence Checking of Quantum Circuits
06/17/2613.2 Equivalence Checking of Quantum Circuits : 240
QMA is the quantum analogue of the classical complexity class NP and, in fact, includes NP.
13.3 Related Work
06/17/2613.3 Related Work : 246
L. Burgholzer, R. Wille, Exploiting reversible computing for verification: Potential, possible paths, and consequences, in ASPDAC (2023
14. Advanced Equivalence Checking for Quantum Circuits
06/17/2614. Advanced Equivalence Checking for Quantum Circuits : 250
Quantum circuits are inherently reversible.
14.1.1 Potential Power of Reversibility
06/17/2614.1.1 Potential Power of Reversibility : 253
14.1.1 Potential Power of Reversibility
14.2.2 Proportional Strategy
06/17/2614.2.2 Proportional Strategy : 262
For the problem at hand, this strategy constitutes near-optimal performance.
14.5 Summary of Results
06/17/2614.5 Summary of Results : 278
arbitrary single-qubit operations combined with CNOT form a universal gate-set [10].
15.2.3 Using Knowledge About the Optimization Step
06/17/2615.2.3 Using Knowledge About the Optimization Step : 288
Qiskit optimization level O0
16.2.2 Local Quantum Stimuli
06/17/2616.2.2 Local Quantum Stimuli : 307
Bloch sphere
17. Equivalence Checking of Quantum Circuits with the ZX-Calculus
07/01/2617. Equivalence Checking of Quantum Circuits with the ZX-Calculus : 322
the ZX-calculus [1–4] (cf. Sect. 3.3) has been mainly used for compilation and optimization of quantum circuits.
07/01/2617. Equivalence Checking of Quantum Circuits with the ZX-Calculus : 322
optimization of quantum circuits
07/01/2617. Equivalence Checking of Quantum Circuits with the ZX-Calculus : 322
in [1], it is briefly mentioned that the optimization algorithm proposed in that paper can also be used to check the equivalence of two quantum circuits.
07/01/2617. Equivalence Checking of Quantum Circuits with the ZX-Calculus : 323
methodology proposed in Chap. 14. One of the circuits is inverted and combined with the remaining circuit
07/01/2617. Equivalence Checking of Quantum Circuits with the ZX-Calculus : 323
(PyZX, Kissinger and van de Wetering [5]) is written in Python, which is known to yield suboptimal performance compared to compiled languages like C++.
17.1 Equivalence Checking of Graph-Like ZX-Diagrams
07/02/2617.1 Equivalence Checking of Graph-Like ZX-Diagrams : 327
In [3], the authors introduce an alternative structure for ZX-diagrams coupled with additional rewrite rules. A ZX-diagram is graph-like when:
1.
All spiders are Z-spiders.
-
Z-spiders are only connected via Hadamard edges.
-
There are no parallel Hadamard edges or self-loops.
-
Every input or output is connected to a Z-spider, and every Z-spider is connected to at most one input or output.
07/02/2617.1 Equivalence Checking of Graph-Like ZX-Diagrams : 327
every ZX-diagram is equal to a graph-like ZX-diagram [3].
07/02/2617.1 Equivalence Checking of Graph-Like ZX-Diagrams : 329
simplification algorithm that reduces every ZX-diagram into a reduced gadget form [4
07/02/2617.1 Equivalence Checking of Graph-Like ZX-Diagrams : 329
in general, there is no unique reduced gadget form for a ZX-diagram.
17.2.1 Handling Inaccuracies
07/02/2617.2.1 Handling Inaccuracies : 330
Clifford+T circuit
T must refer to that sort of gate
07/02/2617.2.1 Handling Inaccuracies : 332
ZX-diagrams are, in essence, tensor networks [8, 9].
17.2.3 Handling Ancillaries
07/02/2617.2.3 Handling Ancillaries : 334
ancillaries are qubits that have a constant initial state and end in the same state
17.3 Completeness
07/02/2617.3 Completeness : 335
the ruleset provided in this paper is complete for circuits solely composed of Clifford gates [10]
07/02/2617.3 Completeness : 337
NP is a subset of QMA
07/02/2617.3 Completeness : 337
a reduced gadget form can be derived in polynomial time with respect to the number of spiders of the original diagram [4]
17.4 Summary of Results
07/02/2617.4 Summary of Results : 339
For circuits with many qubits but fewer gates, the ZX-calculus-based approach performs more favorably
so, cost-aware synthesized circs
07/02/2617.4 Summary of Results : 339
For larger circuits, the problem itself is too complex to be solved for the ZX-calculus-based approach.
07/01/2617.4 Summary of Results : 340
1.J. van de Wetering, ZX-calculus for the working quantum computer scientist (2020). arXiv preprint: arXiv:2012.13966
presents optimization algorithm
07/02/2617.4 Summary of Results : 340
3.R. Duncan, A. Kissinger, S. Perdrix, J. van de Wetering, Graph-theoretic simplification of quantum circuits with the ZX-calculus (2019). arXiv preprint: arXiv:1902.03178
07/02/2617.4 Summary of Results : 341
4.A. Kissinger, J. van de Wetering, Reducing T-count with the ZX-calculus. Phys. Rev. A (2020)
reduced gadget form
interesting Phys. Rev. A published a comp sci paper
07/02/2617.4 Summary of Results : 341
T-count
transistor count?
18.1.1 Running Example
07/02/2618.1.1 Running Example : 346
Quantum Phase Estimation (QPE, Nielsen and Chuang [4]
07/02/2618.1.1 Running Example : 346
quantum principal component analysis [7] for machine learning
18.1.3 Resulting Problem
07/03/2618.1.3 Resulting Problem : 351
any circuit containing non-unitary operations can be transformed to a circuit only containing unitary operations and no intermediate measurements by combining well-known results from quantum information theory
like how activation functions in ML introduce non-linearity?
18.2 Unitary Reconstruction via Circuit Transformation
07/03/2618.2 Unitary Reconstruction via Circuit Transformation : 353
a reset can be interpreted as measuring a qubit, applying an X operation conditioned on the measurement result being |1〉 and then discarding the measurement result. Theoretically, any reset operation can be replaced by introducing a new qubit and applying all subsequent operations involving the qubit to be reset to the new qubit.
This is how initialization to 0 is done.
07/03/2618.2 Unitary Reconstruction via Circuit Transformation : 353
one of the most fundamental results in quantum computing: the deferred measurement principle [4]. This principle states that delaying measurements until the end of a quantum computation does not affect the probability distribution of outcomes
07/03/2618.2 Unitary Reconstruction via Circuit Transformation : 355
The proposed transformation scheme “touches” nothing but reset, measurement, and classically controlled operations—which are “reversed”.
18.3 Extracting Measurement Outcome Distributions via Simulation
07/03/2618.3 Extracting Measurement Outcome Distributions via Simulation : 356
18.3 Extracting Measurement Outcome Distributions via Simulation
very interesting
07/03/2618.3 Extracting Measurement Outcome Distributions via Simulation : 356
we argue that it might be sufficient to show that two realizations of such an algorithm produce the same distribution of measurement probabilities given the fixed input state.
07/07/2618.3 Extracting Measurement Outcome Distributions via Simulation : 358
the following, we propose a technique that allows one to extract the complete set of measurement probabilities for a dynamic circuit given a particular input state. To this end, consider a quantum circuit G involving m measurements. Then each measurement during the circuit simulation constitutes a branching point where the probabilities of the qubit to be measured are check-pointed and the simulation is split into two independent simulations: one assuming that the measurement outcome is |0〉 and the other assuming the outcome is |1〉. Depending on the outcome being |0〉 or |1〉, a subsequent reset operation is translated into a no-op or an X gate, while any classically controlled operation is ignored or applied, respectively. The probability of observing a particular basis state |𝑖〉=|
(
𝑖
𝑚−1⋯
𝑖
0)
2〉 can then be reconstructed from the product of the check-pointed probabilities along the simulation path corresponding to the outcomes
𝑖
0 to
𝑖
𝑚−1.
sounds like DPLL
19. Equivalence Checking of Parameterized Quantum Circuits: Verifying the Compilation ofVariational Quantum Algorithms
07/10/2619. Equivalence Checking of Parameterized Quantum Circuits: Verifying the Compilation ofVariational Quantum Algorithms : 368
NISQ [2] devices. These algorithms use quantum programs as subroutines in a classical optimization routine to solve problems in chemistry [3], finance [4], discrete optimization [5], and more
07/10/2619. Equivalence Checking of Parameterized Quantum Circuits: Verifying the Compilation ofVariational Quantum Algorithms : 369
optimization routine to solve problems in chemistry [3], finance [4], discrete optimization
à le QCi
19.5 Summary of Results
07/10/2619.5 Summary of Results : 388
J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018)
20. Summary of IV
07/13/2620. Summary of IV : 390
the same time, the fact that quantum operations are reversible offers a potential that is not available in classical computing