Classical simulation of noninteracting-fermion quantum circuits
IBM1
TL;DR: It is shown that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant corresponds to a physical model of noninteracting fermions in one dimension.
Abstract: We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [in Proceedings of the 33rd ACM Symposium on the Theory of Computing (2001), p. 114] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits [E. Knill, R. Laflamme, and G. Milburn, Nature (London) 409, 46 (2001)].
Citations
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TL;DR: The Gottesman-Knill theorem, which says that a stabilizer circuit, a quantum circuit consisting solely of controlled-NOT, Hadamard, and phase gates can be simulated efficiently on a classical computer, is improved in several directions.
Abstract: The Gottesman-Knill theorem says that a stabilizer circuit\char22{}that is, a quantum circuit consisting solely of controlled-NOT (CNOT), Hadamard, and phase gates\char22{}can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor of 2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely available program called CHP (CNOT-Hadamard-phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class $\ensuremath{\bigoplus}\mathsf{L}$, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any $n$-qubit stabilizer circuit into a ``canonical form'' that requires at most $O({n}^{2}∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)$ gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.
969 citations
Cites background from "Classical simulation of noninteract..."
...olvable in classical polynomial time.1 However, Valiant’s model has thus far not found any application, although Terhal and DiVincenzo have shown that it applies to a model of noninteracting fermions [31]. There is one class of quantum circuits that is known to be simulable in classical polynomial time, that does not impose any limit on entanglement, and that arises naturally in several applications. ...
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TL;DR: An overview of the field of Variational Quantum Algorithms is presented and strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage are discussed.
Abstract: Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers due to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will likely not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which use a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges, and highlight the exciting prospects for using them to obtain quantum advantage.
842 citations
TL;DR: It is argued that it is nevertheless misleading to view entanglement as a key resource for quantum‐computational power, as it is necessary for any quantum algorithm to offer an exponential speed‐up over classical computation.
Abstract: For any quantum algorithm operating on pure states, we prove that the presence of multipartite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the...
679 citations
Book•
01 Jan 2007
TL;DR: This book discusses quantum algorithms, a quantum model of computation, and algorithms with super-polynomial speed-up, as well as quantum computational complexity theory and lower bounds.
Abstract: Preface 1. Introduction and background 2. Linear algebra and the Dirac notation 3. Qubits and the framework of quantum mechanics 4. A quantum model of computation 5. Superdense coding and quantum teleportation 6. Introductory quantum algorithms 7. Algorithms with super-polynomial speed-up 8. Algorithms based on amplitude amplification 9. Quantum computational complexity theory and lower bounds 10. Quantum error correction Appendices Bibliography Index
637 citations
06 Jun 2011
TL;DR: In this paper, it was shown that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy, and hence the hierarchy collapses to the third level.
Abstract: We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P#P=BPPNP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per(A)|>=√(n!)poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application.This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
606 citations
References
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TL;DR: It is shown that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors and are robust against errors from photon loss and detector inefficiency.
Abstract: Quantum computers promise to increase greatly the efficiency of solving problems such as factoring large integers, combinatorial optimization and quantum physics simulation. One of the greatest challenges now is to implement the basic quantum-computational elements in a physical system and to demonstrate that they can be reliably and scalably controlled. One of the earliest proposals for quantum computation is based on implementing a quantum bit with two optical modes containing one photon. The proposal is appealing because of the ease with which photon interference can be observed. Until now, it suffered from the requirement for non-linear couplings between optical modes containing few photons. Here we show that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. Our methods exploit feedback from photo-detectors and are robust against errors from photon loss and detector inefficiency. The basic elements are accessible to experimental investigation with current technology.
5,236 citations
"Classical simulation of noninteract..." refers background or methods in this paper
...[1] a new class of quantum computations is introduced that are shown to be efficiently simulatable on a classical device....
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...In the quantum circuit we only allow 1-qubit Hadamard transformations, 1- qubit π/2 phase shifts, 1-qubit Pauli-rotations and 2-qubit CNOT gates and furthermore the final measurements are projections in the two eigenspaces of any sequence of Pauli-matrix observables....
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TL;DR: In this article, a condition for boundary Majorana fermions is expressed as a condition on the bulk electron spectrum, which is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3D p-wave superconductor.
Abstract: Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(-L/l0) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a three-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).
3,234 citations
"Classical simulation of noninteract..." refers methods in this paper
...In order to deal with these general interactions, we transform the set of fermion annihilation and creation operators to a new set of Hermitian operators (associated with so called Majorana fermions [10,14]): c2i = ai + a † i , c2i+1 = −i(ai − a†i ), (28) where i = 0, ....
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TL;DR: In this article, a condition for boundary Majorana fermions is expressed as a condition on the bulk electron spectrum, which is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone.
Abstract: Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length $L$ possesses two ground states with an energy difference proportional to $\exp(-L/l_0)$ and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).
2,986 citations
TL;DR: It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations.
2,980 citations
"Classical simulation of noninteract..." refers background in this paper
...The permanent is a much harder object to calculate exactly than the determinant of the fermion case, in fact this has been proved to be an #P -complete problem [17]....
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TL;DR: In this paper, a scheme that realizes controlled interactions between two distant quantum dot spins is proposed, where the effective long-range interaction is mediated by the vacuum field of a high finesse microcavity.
Abstract: The electronic spin degrees of freedom in semiconductors typically have decoherence times that are several orders of magnitude longer than other relevant time scales. A solid-state quantum computer based on localized electron spins as qubits is therefore of potential interest. Here, a scheme that realizes controlled interactions between two distant quantum dot spins is proposed. The effective long-range interaction is mediated by the vacuum field of a high finesse microcavity. By using conduction-band-hole Raman transitions induced by classical laser fields and the cavity-mode, parallel controlled-not operations, and arbitrary single qubit rotations can be realized.
1,702 citations
"Classical simulation of noninteract..." refers background in this paper
...We can identify the nbit computational basis states |x〉 with a state of n fermionic modes, each of which can be occupied, corresponding to 1, or unoccupied, corresponding to 0....
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