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arXiv:2102.08960 · Physical Review Research

Cooper-pair condensates with nonclassical long-range order on quantum devices

LeeAnn M. Sager-Smith, David A. Mazziotti

Published Jan 3, 2022• 17 citations
arXivHTMLPDF

Abstract

An important problem in quantum information is the practical demonstration of nonclassical long-range order on quantum computers. One of the best known examples of a quantum system with nonclassical long-range order is a superconductor. Here we achieve Cooper-like pairing of qubits on a quantum computer, which can be interpreted as superconducting or superfluid states via a Jordan-Wigner mapping. We rigorously confirm the quantum long-range order by measuring the large $O(N)$ eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step toward more complex modeling of phenomena with significant quantum long-range order on quantum computers such as superconductivity and superfluidity.

DOI
10.1103/physrevresearch.4.013003

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An important problem in quantum information is the practical demonstration of nonclassical long-range order on quantum computers. One of the best known examples of a quantum system with nonclassical long-range order is a superconductor. Here we achieve Cooper-like pairing of qubits on a quantum computer, which can be interpreted as superconducting or superfluid states via a Jordan-Wigner mapping. We rigorously confirm the quantum long-range order by measuring the large O(N)O(N)O(N) eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step toward more complex modeling of phenomena with significant quantum long-range order on quantum computers such as superconductivity and superfluidity.
An important problem in quantum information is the practical demonstration of non-classical long-range order on quantum computers. One of the best known examples of a quantum system with non-classical long-range order is a superconductor. Here we achieve Cooper pairing of qubits on a quantum computer to represent superconducting or superfluid states. We rigorously confirm the quantum long-range order by measuring the large O(N)O(N)O(N) eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step towards more complex modeling of superconductivity and superfluidity as well as other phenomena with significant quantum long-range order on quantum computers.
One of the most correlated quantum phenomena achievable is superconductivity. In superconductivity pairs of electrons condense, in a manner analogous to Bose-Einstein condensation, into the same quantum state with their subsequent entanglement permitting frictionless conductivity. Quantum computers have the potential to predict complex, entangled quantum states at non-exponential cost, unlike their classical counterparts, with possible applications to hard problems from physics to economics. Here, using pairs of qubits to represent the electron pairs, we create superconductivity on a transmon-qubit quantum computer. A series of maximally entangled superconducting states with different numbers of electron pairs is prepared, and then the occupation of the superconducting state is measured by tomography. Each occupation on the order of the number of electrons O(N)O(N)O(N) confirms the formation of a maximally entangled superconducting state. Moreover, the large eigenvalue provides experimental validation of C. N. Yang's derivation of large occupation being associated with off-diagonal long-range order. The work provides a critically important step in predicting strongly entangled quantum physics as well as mathematically related non-local phenomena in the pure and social sciences on quantum computers.
Electrons in a superconductor form Cooper pairs and thus condense into a ground state which can be described by a macroscopic quantum wavefunction. The phase of the wavefunction represents a collective movement of millions of Cooper pairs. Using devices based on Josephson tunnel junctions we are able to control and detect the phase and number of Cooper pairs in a superconductor. Josephson devices are therefore excellent systems for demonstrating macroscopic quantum phenomena. We present a short review of the macroscopic quantum phenomena in superconducting Josephson devices, as well as their applications and prospects in quantum computationl.
Charge ordering is often found in the phase diagram of unconventional superconductors in close proximity to the superconducting state. This has led to the suggestion that fluctuations of charge order can mediate superconducting pairing. While several mechanisms can lead to charge order, one common mechanism is the long-range Coulomb interaction, resulting in a Wigner crystal charge ordered state. For an electron density of 0.5 per site, we investigate the extended Hubbard model on the two-dimensional square lattice using exact diagonalization and density matrix renormalization group methods. Our results show that the strength of pairing decreases with the nearest-neighbor Coulomb interaction strength VVV and remains weaker than the tendency of pairing for noninteracting electrons.
It is known that the loss of phase coherence of Cooper pairs in two-dimensional superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D XY model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analog, various phase transitions can be accurately determined. Below the BKT phase transition, an interlayer Ising long-range order is established at T_{Ising}, and the phase coherence of both intralayers and interlayers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multicritical point. For two inequivalent layers, however, there emerges an intermediate quasi-long-range order phase (T_{Ising}<T<T_{BKT}), where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. So our study provides a promising way to realize the charge-4e superconductivity in a bilayer system.
At any finite temperature, the correlation function $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathcal{r})\ensuremath{\psi}(0)〉$ of a two-dimensional Bose system is known to approach zero as the separation r\mathcal{r}r approaches infinity. This paper is devoted to supporting the point of view that superfluid phenomena may still persist in such a system. To describe the low-temperature behavior of a condensed two-dimensional Bose system, we use a coherent-state representation of the low-lying phonon excitations. The expectation of the particle field operator $〈\ensuremath{\psi}(\mathcal{r})〉$ in a coherent phonon state of the system is shown to be a well-defined function. This function has only small fluctuations in its magnitude and, therefore, it is possible to make the usual arguments for superfluid behavior based on the existence of the condensate wave function. The large phase fluctuations of the function, when the coherentphonon states are collected into the thermal ensemble, give rise to the spatial decay of the correlation function or reduced density matrix $〈{\ensuremath{\psi}}^{+}(\mathcal{r})\ensuremath{\psi}(0)〉$. It is concluded that two-dimensional Bose systems are superfluid at sufficiently low temperatures, even in the inifinite-volume limit. An analogous argument would indicate the possibility of superconductivity in a two-dimensional Fermi system.