arXiv subjects

Mathias Soeken

Publications and source records attributed to Mathias Soeken.

29 records · Page 2Linked to original sources

arXiv2019

Reversible Pebbling Game for Quantum Memory Management

Quantum memory management is becoming a pressing problem, especially given the recent research effort to develop new and more complex quantum algorithms. The only existing automatic method for quantum states clean-up relies on the availability of many extra resources. In this work, we propose an automatic tool for quantum memory management. We show how this problem exactly matches the reversible pebbling game. Based on that, we develop a SAT-based algorithm that returns a valid clean-up strategy, taking the limitations of the quantum hardware into account. The developed tool empowers the designer with the flexibility required to explore the trade-off between memory resources and number of operations. We present three show-cases to prove the validity of our approach. First, we apply the algorithm to straight-line programs, widely used in cryptographic applications. Second, we perform a comparison with the existing approach, showing an average improvement of 52.77%. Finally, we show the advantage of using the tool when synthesizing a quantum circuit on a constrained near-term quantum device.

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arXiv2018

Quantum circuits for floating-point arithmetic

Quantum algorithms to solve practical problems in quantum chemistry, materials science, and matrix inversion often involve a significant amount of arithmetic operations which act on a superposition of inputs. These have to be compiled to a set of fault-tolerant low-level operations and throughout this translation process, the compiler aims to come close to the Pareto-optimal front between the number of required qubits and the depth of the resulting circuit. In this paper, we provide quantum circuits for floating-point addition and multiplication which we find using two vastly different approaches. The first approach is to automatically generate circuits from classical Verilog implementations using synthesis tools and the second is to generate and optimize these circuits by hand. We compare our two approaches and provide evidence that floating-point arithmetic is a viable candidate for use in quantum computing, at least for typical scientific applications, where addition operations usually do not dominate the computation. All our circuits were constructed and tested using the software tools LIQ$Ui|\rangle{}$ and RevKit.

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arXiv2018

Programming Quantum Computers Using Design Automation

Recent developments in quantum hardware indicate that systems featuring more than 50 physical qubits are within reach. At this scale, classical simulation will no longer be feasible and there is a possibility that such quantum devices may outperform even classical supercomputers at certain tasks. With the rapid growth of qubit numbers and coherence times comes the increasingly difficult challenge of quantum program compilation. This entails the translation of a high-level description of a quantum algorithm to hardware-specific low-level operations which can be carried out by the quantum device. Some parts of the calculation may still be performed manually due to the lack of efficient methods. This, in turn, may lead to a design gap, which will prevent the programming of a quantum computer. In this paper, we discuss the challenges in fully-automatic quantum compilation. We motivate directions for future research to tackle these challenges. Yet, with the algorithms and approaches that exist today, we demonstrate how to automatically perform the quantum programming flow from algorithm to a physical quantum computer for a simple algorithmic benchmark, namely the hidden shift problem. We present and use two tool flows which invoke RevKit. One which is based on ProjectQ and which targets the IBM Quantum Experience or a local simulator, and one which is based on Microsoft's quantum programming language Q$\#$.

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arXiv2017

Logic Synthesis for Quantum Computing

We present a synthesis framework to map logic networks into quantum circuits for quantum computing. The synthesis framework is based on LUT networks (lookup-table networks), which play a key role in conventional logic synthesis. Establishing a connection between LUTs in a LUT network and reversible single-target gates in a reversible network allows us to bridge conventional logic synthesis with logic synthesis for quantum computing, despite several fundamental differences. We call our synthesis framework LUT-based Hierarchical Reversible Logic Synthesis (LHRS). Input to LHRS is a classical logic network; output is a quantum network (realized in terms of Clifford+$T$ gates). The framework offers to trade-off the number of qubits for the number of quantum gates. In a first step, an initial network is derived that only consists of single-target gates and already completely determines the number of qubits in the final quantum network. Different methods are then used to map each single-target gate into Clifford+$T$ gates, while aiming at optimally using available resources. We demonstrate the effectiveness of our method in automatically synthesizing IEEE compliant floating point networks up to double precision. As many quantum algorithms target scientific simulation applications, they can make rich use of floating point arithmetic components. But due to the lack of quantum circuit descriptions for those components, it can be difficult to find a realistic cost estimation for the algorithms. Our synthesized benchmarks provide cost estimates that allow quantum algorithm designers to provide the first complete cost estimates for a host of quantum algorithms. Thus, the benchmarks and, more generally, the LHRS framework are an essential step towards the goal of understanding which quantum algorithms will be practical in the first generations of quantum computers.

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arXiv2016

Design Automation and Design Space Exploration for Quantum Computers

A major hurdle to the deployment of quantum linear systems algorithms and recent quantum simulation algorithms lies in the difficulty to find inexpensive reversible circuits for arithmetic using existing hand coded methods. Motivated by recent advances in reversible logic synthesis, we synthesize arithmetic circuits using classical design automation flows and tools. The combination of classical and reversible logic synthesis enables the automatic design of large components in reversible logic starting from well-known hardware description languages such as Verilog. As a prototype example for our approach we automatically generate high quality networks for the reciprocal $1/x$, which is necessary for quantum linear systems algorithms.

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arXiv2015

Translating between the roots of the identity in quantum computers

The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $\langle H, T\rangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or also the eighth root of the identity $I$. The Hadamard matrix $H$ can be used to translate between the Pauli matrices, since $(HTH)^4$ gives Pauli $X$. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.

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arXiv2015

Self-Inverse Functions and Palindromic Circuits

We investigate the subclass of reversible functions that are self-inverse and relate them to reversible circuits that are equal to their reverse circuit, which are called palindromic circuits. We precisely determine which self-inverse functions can be realized as a palindromic circuit. For those functions that cannot be realized as a palindromic circuit, we find alternative palindromic representations that require an extra circuit line or quantum gates in their construction. Our analyses make use of involutions in the symmetric group $S_{2^n}$ which are isomorphic to self-inverse reversible function on $n$ variables.

cs.ET↗

arXiv2014

Ancilla-free synthesis of large reversible functions using binary decision diagrams

The synthesis of reversible functions has been an intensively studied research area in the last decade. Since almost all proposed approaches rely on representations of exponential size (such as truth tables and permutations), they cannot be applied efficiently to reversible functions with more than 15 variables. In this paper, we propose an ancilla-free synthesis approach based on Young subgroups using symbolic function representations that can efficiently be implemented with binary decision diagrams (BDDs). As a result, the algorithm not only allows to synthesize large reversible functions without adding extra lines, called ancilla, but also leads to significantly smaller circuits compared to existing approaches.

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arXiv2014

Embedding of Large Boolean Functions for Reversible Logic

Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function before they can be synthesized. Thus far, an optimal embedding is guaranteed only for small functions, whereas a significant overhead results when large functions are considered. In this paper, we study this issue. We prove that determining an optimal embedding is coNP-hard already for restricted cases. Then, we propose heuristic and exact methods for determining both the number of additional lines as well as a corresponding embedding. For the approaches we considered sums of products and binary decision diagrams as function representations. Experimental evaluations show the applicability of the approaches for large functions. Consequently, the reversible embedding of large functions is enabled as a precursor to subsequent synthesis.

cs.ET↗

arXiv2014

A framework for reversible circuit complexity

Reversible single-target gates are a generalization of Toffoli gates which are a helpful formal representation for the description of synthesis algorithms but are too general for an actual implementation based on some technology. There is an exponential lower bound on the number of Toffoli gates required to implement any reversible function, however, there is also a linear upper bound on the number of single-target gates which can be proven using a constructive proof based on a former presented synthesis algorithm. Since single-target gates can be mapped to a cascade of Toffoli gates, this synthesis algorithm provides an interesting framework for reversible circuit complexity. The paper motivates this framework and illustrates first possible applications based on it.

cs.ET↗

arXiv2013

On quantum circuits employing roots of the Pauli matrices

The Pauli matrices are a set of three 2x2 complex Hermitian, unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum circuits. Techniques for simplifying such circuits are given. In particular, we show how those techniques can be used to find a circuit of Clifford+T gates starting from a circuit composed of gates from the well studied NCV library.

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