Adiabatic preparation of topological order

Topological order characterizes those phases of matter that defy a description in terms of symmetry and cannot be distinguished in terms of local order parameters. Here we show that a system of n spins forming a lattice on a Riemann surface can undergo a second order quantum phase transition between a spin-polarized phase and a string-net condensed phase. This is an example of a quantum phase transition between magnetic and topological order. We furthermore show how to prepare the topologically ordered phase through adiabatic evolution in a time that is upper bounded by O(root n). This provides a physically plausible method for constructing and initializing a topological quantum memory. RI Lidar, Daniel/A-5871-200

Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap. RI Lidar, Daniel/A-5871-200

Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order

We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-1/2 particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point. RI Haas, Stephan/C-4103-2008; Lidar, Daniel/A-5871-200

Quantum Adiabatic Brachistochrone

We formulate a time-optimal approach to adiabatic quantum computation (AQC). A corresponding natural Riemannian metric is also derived, through which AQC can be understood as the problem of finding a geodesic on the manifold of control parameters. This geometrization of AQC is demonstrated through two examples, where we show that it leads to improved performance of AQC, and sheds light on the roles of entanglement and curvature of the control manifold in algorithmic performance. RI Lidar, Daniel/A-5871-200

Universal Quantum Computation with the Exchange Interaction

Experimental implementations of quantum computer architectures are now being investigated in many different physical settings. The full set of requirements that must be met to make quantum computing a reality in the laboratory [1] is daunting, involving capabilities well beyond the present state of the art. In this report we develop a significant simplification of these requirements that can be applied in many recent solid-state approaches, using quantum dots [2], and using donor-atom nuclear spins [3] or electron spins [4]. In these approaches, the basic two-qubit quantum gate is generated by a tunable Heisenberg interaction (the Hamiltonian is $H_{ij}=J(t){\vec S}_i\cdot{\vec S}_j$ between spins $i$ and $j$), while the one-qubit gates require the control of a local Zeeman field. Compared to the Heisenberg operation, the one-qubit operations are significantly slower and require substantially greater materials and device complexity, which may also contribute to increasing the decoherence rate. Here we introduce an explicit scheme in which the Heisenberg interaction alone suffices to exactly implement any quantum computer circuit, at a price of a factor of three in additional qubits and about a factor of ten in additional two-qubit operations. Even at this cost, the ability to eliminate the complexity of one-qubit operations should accelerate progress towards these solid-state implementations of quantum computation.Comment: revtex, 2 figures, this version appeared in Natur

Decoherence induced deformation of the ground state in adiabatic quantum computation

Despite more than a decade of research on adiabatic quantum computation (AQC), its decoherence properties are still poorly understood. Many theoretical works have suggested that AQC is more robust against decoherence, but a quantitative relation between its performance and the qubits' coherence properties, such as decoherence time, is still lacking. While the thermal excitations are known to be important sources of errors, they are predominantly dependent on temperature but rather insensitive to the qubits' coherence. Less understood is the role of virtual excitations, which can also reduce the ground state probability even at zero temperature. Here, we introduce normalized ground state fidelity as a measure of the decoherence-induced deformation of the ground state due to virtual transitions. We calculate the normalized fidelity perturbatively at finite temperatures and discuss its relation to the qubits' relaxation and dephasing times, as well as its projected scaling properties.Comment: 10 pages, 3 figure

'Designer atoms' for quantum metrology

Entanglement is recognized as a key resource for quantum computation and quantum cryptography. For quantum metrology, the use of entangled states has been discussed and demonstrated as a means of improving the signal-to-noise ratio. In addition, entangled states have been used in experiments for efficient quantum state detection and for the measurement of scattering lengths. In quantum information processing, manipulation of individual quantum bits allows for the tailored design of specific states that are insensitive to the detrimental influences of an environment. Such 'decoherence-free subspaces' protect quantum information and yield significantly enhanced coherence times. Here we use a decoherence-free subspace with specifically designed entangled states to demonstrate precision spectroscopy of a pair of trapped Ca+ ions; we obtain the electric quadrupole moment, which is of use for frequency standard applications. We find that entangled states are not only useful for enhancing the signal-to-noise ratio in frequency measurements - a suitably designed pair of atoms also allows clock measurements in the presence of strong technical noise. Our technique makes explicit use of non-locality as an entanglement property and provides an approach for 'designed' quantum metrology

Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization

One fundamental requirement for quantum computation is to perform universal manipulations of quantum bits at rates much faster than the qubit's rate of decoherence. Recently, fast gate operations have been demonstrated in logical spin qubits composed of two electron spins where the rapid exchange of the two electrons permits electrically controllable rotations around one axis of the qubit. However, universal control of the qubit requires arbitrary rotations around at least two axes. Here we show that by subjecting each electron spin to a magnetic field of different magnitude we achieve full quantum control of the two-electron logical spin qubit with nanosecond operation times. Using a single device, a magnetic field gradient of several hundred milliTesla is generated and sustained using dynamic nuclear polarization of the underlying Ga and As nuclei. Universal control of the two-electron qubit is then demonstrated using quantum state tomography. The presented technique provides the basis for single and potentially multiple qubit operations with gate times that approach the threshold required for quantum error correction.Comment: 11 pages, 4 figures. Supplementary Material included as ancillary fil

Quantum adiabatic machine learning

We develop an approach to machine learning and anomaly detection via quantum adiabatic evolution. In the training phase we identify an optimal set of weak classifiers, to form a single strong classifier. In the testing phase we adiabatically evolve one or more strong classifiers on a superposition of inputs in order to find certain anomalous elements in the classification space. Both the training and testing phases are executed via quantum adiabatic evolution. We apply and illustrate this approach in detail to the problem of software verification and validation.Comment: 21 pages, 9 figure