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IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

Overcoming the Tradeoff Between Accuracy and Compactness in Decision Diagrams for Quantum Computation

Philipp Niemann, Alwin Zulehner, Rolf Drechsler, Robert Wille

Published Mar 3, 2020• 19 citations
Publisher

Abstract

Quantum computation promises to solve many hard or infeasible problems substantially faster than classical solutions. The involvement of big players like Google, IBM, Intel, Rigetti, or Microsoft furthermore led to a momentum which increases the demand for automated design methods for quantum computations. In this context, decision diagrams for quantum computation provide a major pillar as they allow to efficiently represent quantum states and quantum operations which, otherwise, have to be described in terms of exponentially large state vectors and unitary matrices. However, current decision diagrams for the quantum domain suffer from a tradeoff between accuracy and compactness, since: 1) small errors that are inevitably introduced by the limited precision of floating-point arithmetic can harm the compactness (i.e., the size of the decision diagram) significantly and 2) overcompensating these errors (to increase compactness) may lead to an information loss and introduces numerical instabilities. In this article, we describe and evaluate the effects of this tradeoff which clearly motivates the need for a solution that is perfectly accurate and compact at the same time. More precisely, we show that the tradeoff indeed weakens current design automation approaches for quantum computation (possibly leading to corrupted results or infeasible run-times). To overcome this, we propose an alternative approach that utilizes an algebraic representation of the occurring complex and irrational numbers and outline how this can be incorporated in a decision diagram which is suited for quantum computation. Evaluations show that-at the cost of an overhead which is moderate in many cases-the proposed algebraic solution indeed overcomes the tradeoff between accuracy and compactness that is present in current numerical solutions.

DOI
10.1109/tcad.2020.2977603

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A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and $T$ gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of $T$ gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as $2^t \cdot poly(g, n)$, where $t$ ($g$) is the number of $t$ gates (Clifford gates) and $n$ the number of qubits. Our proofs are based on a $T$-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+$T$ gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.

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Quantum computation promises to solve many hard or infeasible problems substantially faster than classical solutions. The involvement of big players like Google, IBM, Intel, Rigetti, or Microsoft furthermore led to a momentum which increases the demand for automated design methods for quantum computations. In this context, decision diagrams for quantum computation provide a major pillar as they allow to efficiently represent quantum states and quantum operations which, otherwise, have to be described in terms of exponentially large state vectors and unitary matrices. However, current decision diagrams for the quantum domain suffer from a tradeoff between accuracy and compactness, since: 1) small errors that are inevitably introduced by the limited precision of floating-point arithmetic can harm the compactness (i.e., the size of the decision diagram) significantly and 2) overcompensating these errors (to increase compactness) may lead to an information loss and introduces numerical instabilities. In this article, we describe and evaluate the effects of this tradeoff which clearly motivates the need for a solution that is perfectly accurate and compact at the same time. More precisely, we show that the tradeoff indeed weakens current design automation approaches for quantum computation (possibly leading to corrupted results or infeasible run-times). To overcome this, we propose an alternative approach that utilizes an algebraic representation of the occurring complex and irrational numbers and outline how this can be incorporated in a decision diagram which is suited for quantum computation. Evaluations show that-at the cost of an overhead which is moderate in many cases-the proposed algebraic solution indeed overcomes the tradeoff between accuracy and compactness that is present in current numerical solutions.
Quantum computation is a promising research field since it allows to conduct certain tasks exponentially faster than on conventional machines. As in the conventional domain, decision diagrams are heavily used in different design tasks for quantum computation like synthesis, verification, or simulation. However, unlike decision diagrams for the conventional domain, decision diagrams for quantum computation as of now suffer from a trade-off between accuracy and compactness that requires parameter fine-tuning on a case-by-case basis. In this work, we-for the first time-describe and evaluate the effects of this trade-off. Moreover, we propose an alternative approach that utilizes an algebraic representation of the occurring irrational numbers and outline how this can be incorporated in a decision diagram in order to overcome this trade-off.
Quantum computers promise to solve important problems faster than conventional computers ever could. Underneath is a fundamentally different computational primitive that introduces new challenges for the development of software tools that aid designers of corresponding quantum algorithms. The different computational primitives render classical simulation of quantum circuits particularly challenging. While the logic simulation of conventional circuits is comparatively simple with linear complexity with respect to the number of gates, quantum circuit simulation has to deal with the exponential memory requirements to represent quantum states on non-quantum hardware with respect to the number of qubits. Decision Diagrams (DDs) address this challenge through exploitation of redundancies in matrices and vectors to provide significantly more compact representations in many cases. Moreover, the probabilistic nature of quantum computations enables another angle to tackle the complexity: Quantum algorithms are resistant to some degree against small inaccuracies in the quantum state as these only lead to small changes in the outcome probabilities. We propose to exploit this resistance against (small) errors to gain even more compact decision diagrams. In this work, we investigate the potential of approximation in quantum circuit simulation in detail. To this end, we first present four dedicated schemes that exploit the error resistance and efficiently approximate quantum states represented by decision diagrams. Subsequently, we propose two simulation strategies that utilize those approximations schemes in order to improve the efficiency of DD-based quantum circuit simulation, while, at the same time, allowing the user to control the resulting degradation in accuracy. We empirically show that the proposed approximation schemes reduce the size of decision diagrams substantially and also analytically prove the effect of multiple approximations on the attained accuracy. Eventually, this enables speed-ups of the resulting approximate quantum circuit simulation of up to several orders of magnitudes—again, while controlling the fidelity of the result.
Quantum computing promises substantial speedups by exploiting quantum mechanical phenomena such as superposition and entanglement. Corresponding design methods require efficient means of representation and manipulation of quantum functionality. In the classical domain, decision diagrams have been successfully employed as a powerful alternative to straightforward means such as truth tables. This motivated extensive research on whether decision diagrams provide similar potential in the quantum domain-resulting in new types of decision diagrams capable of substantially reducing the complexity of representing quantum states and functionality. From an implementation perspective, many concepts and techniques from the classical domain can be re-used in order to implement decision diagrams packages for the quantum realm. However, new problems-namely how to efficiently handle complex numbers-arise. In this work, we propose a solution to overcome these problems. Experimental evaluations confirm that this yields improvements of orders of magnitude in the runtime needed to create and to utilize these decision diagrams. The resulting implementation is publicly available as a quantum DD package at http://iic.jku.at/eda/research/quantum_dd.
Quantum computers promise to solve important problems faster than conventional computers. However, unleashing this power has been challenging. In particular, design automation runs into (1) the probabilistic nature of quantum computation and (2) exponential requirements for computational resources on non-quantum hardware. In quantum circuit simulation, Decision Diagrams (DDs) have previously shown to reduce the required memory in many important cases by exploiting redundancies in the quantum state. In this paper, we show that this reduction can be amplified by exploiting the probabilistic nature of quantum computers to achieve even more compact representations. Specifically, we propose two new DD-based simulation strategies that approximate the quantum states to attain more compact representations, while, at the same time, allowing the user to control the resulting degradation in accuracy. We also analytically prove the effect of multiple approximations on the attained accuracy and empirically show that the resulting simulation scheme enables speed-ups up to several orders of magnitudes.
Classical representations of quantum states and operations as vectors and matrices are plagued by an exponential growth in memory and runtime requirements for increasing system sizes. Based on their use in classical computing, an alternative data structure known as Decision Diagrams (DDs) has been proposed, which, in many cases, provides both a more compact representation and more efficient computation. In the classical realm, decades of research have been conducted on DDs and numerous variations tailored for specific applications exist. However, DDs for quantum computing are just in their infancy and there is still room for tailoring them to this new technology. In particular, existing representations of DDs require extending all operations in a quantum circuit to the full system size through extension by nodes representing identity matrices. In this work, we make an important step forward for quantum DDs by stripping these identity structures from quantum operations. This significantly reduces the number of nodes required to represent them as well as eases the pressure on key building blocks of their implementation. As a result, we obtain a structure that is more natural for quantum computing and significantly speeds up computations—with a runtime improvement of up to 70 × compared to the state-of-the-art.
Exact quantum decision diagrams with scaling guarantees for Clifford+TTT circuits and beyond
A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and TTT gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of TTT gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as 2t⋅poly(g,n)2^t \cdot poly(g, n)2t⋅poly(g,n), where ttt (ggg) is the number of ttt gates (Clifford gates) and nnn the number of qubits. Our proofs are based on a TTT-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+TTT gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.