Experimental implementation of a four-player quantum game

Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now only two-player quantum games have been demonstrated. Here we report the first experiment that implements a four-player quantum Minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate quantum player strategies give equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our result demonstrate for the first time how non-trivial equilibria can arise in a competitive situation involving quantum agents and pave the way for a range of quantum transaction applications.Comment: 9 pages, 5 figure

Towards practical classical processing for the surface code: timing analysis

Topological quantum error correction codes have high thresholds and are well suited to physical implementation. The minimum weight perfect matching algorithm can be used to efficiently handle errors in such codes. We perform a timing analysis of our current implementation of the minimum weight perfect matching algorithm. Our implementation performs the classical processing associated with an nxn lattice of qubits realizing a square surface code storing a single logical qubit of information in a fault-tolerant manner. We empirically demonstrate that our implementation requires only O(n^2) average time per round of error correction for code distances ranging from 4 to 512 and a range of depolarizing error rates. We also describe tests we have performed to verify that it always obtains a true minimum weight perfect matching.Comment: 13 pages, 13 figures, version accepted for publicatio

Modal expansions and non-perturbative quantum field theory in Minkowski space

We introduce a spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1+1 dimensional phi^4 theory, identifying both one-particle and multi-particle contributions. We discuss the computational limits of existing diagonalization algorithms and suggest new quasi-sparse eigenvector methods to handle very large Fock spaces and higher dimensional field theories.Comment: new material added, 12 pages, 6 figure

Donor Electron Wave Functions for Phosphorus in Silicon: Beyond Effective Mass Theory

We calculate the electronic wave-function for a phosphorus donor in silicon by numerical diagonalisation of the donor Hamiltonian in the basis of the pure crystal Bloch functions. The Hamiltonian is calculated at discrete points localised around the conduction band minima in the reciprocal lattice space. Such a technique goes beyond the approximations inherent in the effective-mass theory, and can be modified to include the effects of altered donor impurity potentials, externally applied electro-static potentials, as well as the effects of lattice strain. Modification of the donor impurity potential allows the experimentally known low-lying energy spectrum to be reproduced with good agreement, as well as the calculation of the donor wavefunction, which can then be used to calculate parameters important to quantum computing applications.Comment: 10 pages, 5 figure

Asymmetric quantum error correction via code conversion

In many physical systems it is expected that environmental decoherence will exhibit an asymmetry between dephasing and relaxation that may result in qubits experiencing discrete phase errors more frequently than discrete bit errors. In the presence of such an error asymmetry, an appropriately asymmetric quantum code - that is, a code that can correct more phase errors than bit errors - will be more efficient than a traditional, symmetric quantum code. Here we construct fault tolerant circuits to convert between an asymmetric subsystem code and a symmetric subsystem code. We show that, for a moderate error asymmetry, the failure rate of a logical circuit can be reduced by using a combined symmetric asymmetric system and that doing so does not preclude universality.Comment: 5 pages, 8 figures, presentation revised, figures and references adde

Dopant metrology in advanced FinFETs

Ultra-scaled FinFET transistors bear unique fingerprint-like device-to-device differences attributed to random single impurities. This paper describes how, through correlation of experimental data with multimillion atom tight-binding simulations using the NEMO 3-D code, it is possible to identify the impurity's chemical species and determine their concentration, local electric field and depth below the Si/SiO$_{\mathrm{2}}$ interface. The ability to model the excited states rather than just the ground state is the critical component of the analysis and allows the demonstration of a new approach to atomistic impurity metrology.Comment: 6 pages, 3 figure

Engineered valley-orbit splittings in quantum confined nanostructures in silicon

An important challenge in silicon quantum electronics in the few electron regime is the potentially small energy gap between the ground and excited orbital states in 3D quantum confined nanostructures due to the multiple valley degeneracies of the conduction band present in silicon. Understanding the "valley-orbit" (VO) gap is essential for silicon qubits, as a large VO gap prevents leakage of the qubit states into a higher dimensional Hilbert space. The VO gap varies considerably depending on quantum confinement, and can be engineered by external electric fields. In this work we investigate VO splitting experimentally and theoretically in a range of confinement regimes. We report measurements of the VO splitting in silicon quantum dot and donor devices through excited state transport spectroscopy. These results are underpinned by large-scale atomistic tight-binding calculations involving over 1 million atoms to compute VO splittings as functions of electric fields, donor depths, and surface disorder. The results provide a comprehensive picture of the range of VO splittings that can be achieved through quantum engineering.Comment: 4 pages, 4 figure

Atomistic simulations of adiabatic coherent electron transport in triple donor systems

A solid-state analogue of Stimulated Raman Adiabatic Passage can be implemented in a triple well solid-state system to coherently transport an electron across the wells with exponentially suppressed occupation in the central well at any point of time. Termed coherent tunneling adiabatic passage (CTAP), this method provides a robust way to transfer quantum information encoded in the electronic spin across a chain of quantum dots or donors. Using large scale atomistic tight-binding simulations involving over 3.5 million atoms, we verify the existence of a CTAP pathway in a realistic solid-state system: gated triple donors in silicon. Realistic gate profiles from commercial tools were combined with tight-binding methods to simulate gate control of the donor to donor tunnel barriers in the presence of cross-talk. As CTAP is an adiabatic protocol, it can be analyzed by solving the time independent problem at various stages of the pulse - justifying the use of time-independent tight-binding methods to this problem. Our results show that a three donor CTAP transfer, with inter-donor spacing of 15 nm can occur on timescales greater than 23 ps, well within experimentally accessible regimes. The method not only provides a tool to guide future CTAP experiments, but also illuminates the possibility of system engineering to enhance control and transfer times.Comment: 8 pages, 5 figure

A precise CNOT gate in the presence of large fabrication induced variations of the exchange interaction strength

We demonstrate how using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate in a suitably characterised system means concatenation of the composite pulse is unnecessary, hence reducing operation time, and ensuring the gate operates below the threshold required for fault-tolerant quantum computation.Comment: 9 pages, 8 figure