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arXiv:2011.01239

Supersymmetry and Quantum Computation

P. Marcos Crichigno

Published Nov 2, 2020• 3 citations
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Abstract

The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of $\mathcal N \leq 2$ quantum mechanical systems is $\#P$-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.

DOI
10.48550/arxiv.2011.01239

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The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of N≤2\mathcal N \leq 2N≤2 quantum mechanical systems is #P\#P#P-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1\mathcal{N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2\mathcal{N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
Quantum algorithms promise up to a superpolynomial factor speedup in computational tasks compared to their classical counterparts. Quantum computers are now approaching sizes where quantum algorithms can be demonstrated. As a result, the implementation of algorithms as quantum circuits has become an important research topic. In this work, we provide an overview of quantum computation and the design of quantum circuits. Properties unique to quantum computation are outlined. We introduce how a quantum algorithm is translated into a circuit and show measures used to assess the performance of quantum circuits.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1{\cal N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2{\cal N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.