# Qubit-efficient encoding schemes for binary optimisation problems

Benjamin Tan1, Marc-Antoine Lemonde1, Supanut Thanasilp1, Jirawat Tangpanitanon1, and Dimitris G. Angelakis1,2

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
2School of Electrical and Computer Engineering, Technical University of Crete, Chania, Greece 73100

### Abstract

We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of $n_c$ classical variables can be implemented on $\mathcal O(\log n_c)$ number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of $64$ classical variables using $7$ qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a $42$-variable Max-Cut problem using only $8$ qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.

One promising application where quantum computers are expected to become a disruptive innovation is in the field of combinatorial optimization. So far, most quantum algorithms for combinatorial problems entail mapping each classical variable to single qubit in the quantum device. While this mapping in principle allows for an efficient search over the exponentially large number of possible solutions, the quantum resources required and restricted connectivity of current quantum hardware limits the problem sizes to toy models. Current state-of-the-art quantum experiments have found approximate solutions for a fully connected problem of only $17$ variables while modern classical methods can find solutions for problem sizes with over $10^4$ variables. In our work, we introduce an alternative encoding scheme that could allow intermediate-scale quantum devices to tackle problem sizes rivaling the biggest instances that can be solved using modern classical methods.

We adopt a fundamentally different approach by using a quantum state to encode correlations between restricted subsets of variables. In the simplest allowed encoding, each subset consists of a single variable, allowing for approximate solutions to be found using exponentially fewer qubits. The flexibility of our encoding scheme allows for additional qubits to systematically increase the correlations captured by the quantum state. Using numerical simulations, we demonstrate this encoding scheme on a $64$-variable optimization problem using only $7$ qubits and $104$ gates in the limiting case where no correlations are captured. We also demonstrate, with a $42$-variable example using $8$ qubits, how capturing two-body correlation yields better results. Lastly, we include results showing how this reduction in resources is advantageous compared to standard quantum approaches when implemented on noisy quantum hardware.

Moving forward, we aim to explore the possibilities of enhancing the performance of this encoding scheme to compete alongside state-of-the-art classical algorithms

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### Cited by

[1] Adam Glos, Aleksandra Krawiec, and Zoltán Zimborás, "Space-efficient binary optimization for variational computing", arXiv:2009.07309.

[2] V. E. Zobov and I. S. Pichkovskiy, "Clustering by quantum annealing on three-level quantum elements qutrits", arXiv:2102.09205.

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