Session D29: Quantum Computing and Simulation I

Scalable Quantum Computing Over the Rainbow

The physical implementation of nontrivial quantum computing is an experimental challenge due to decoherence and the need for scalability. Recently we proved a novel theoretical scheme for realizing a scalable quantum register of very large size, entangled in a cluster state, in the optical frequency comb (OFC) defined by the eigenmodes of a single optical parametric oscillator (OPO). The classical OFC is well known as implemented by the femtosecond, carrier-envelope-phase- and mode-locked lasers which have redefined frequency metrology in recent years. The quantum OFC is a set of harmonic oscillators, or Qmodes, whose amplitude and phase quadratures are continuous variables, the manipulation of which is a mature field for one or two Qmodes. We have shown that the nonlinear optical medium of a single OPO can be engineered, in a sophisticated but already demonstrated manner, so as to entangle in constant time the OPO's OFC into a finitely squeezed, Gaussian cluster state suitable for universal quantum computing over continuous variables. Here we summarize our theoretical result and survey the ongoing experimental efforts in this direction.   

Implementing quantum gates through scattering between a static and a flying qubit

We investigate whether a two-qubit quantum gate can be implemented in a scattering process involving a flying and a static qubit. We focus on a paradigmatic setup made out of a mobile particle and a quantum impurity, whose respective spin degrees of freedom couple to each other during a one-dimensional scattering process. A condition for the occurrence of quantum gates is derived in terms of spin-dependent transmission coefficients. This can be fulfilled through the insertion of an additional narrow potential barrier. Under resonance conditions this procedure enables a gate only for Heisenberg interactions and fails for an XY interaction. We show the existence of parameter regimes for which gates able to establish a maximum amount of entanglement can be implemented. The gates are found to be robust to variations of the optimal parameters.   

The 2D AKLT state is a universal quantum computational resource

We demonstrate that the two-dimensional AKLT state on a honeycomb lattice is a universal resource for measurement-based quantum computation. Our argument proceeds by reduction of the AKLT state to a 2D cluster state, which is already known to be universal, and consists of two steps. First, we devise a local POVM by which the AKLT state is mapped to a random 2D graph state. Second, we show by Monte Carlo simulations that the connectivity properties of these random graphs are governed by percolation, and that typical graphs are in the connected phase. The corresponding graph states can then be transformed to 2D cluster states.   

Robust benchmarking of quantum processes

Fault-tolerant threshold theorems show that as long as the noise affecting a quantum system is below some threshold, reliable quantum computation can take place. As a result, methods for noise characterization are of great interest in quantum information theory. Unfortunately, methods for complete noise characterization scale exponentially in the number of qubits comprising the system. This non-scalability highlights the importance of developing mathematical methods for scalable partial characterization of the noise affecting a quantum system. We discuss a randomized benchmarking protocol that provides fitting models for the fidelity decay of the noisy gates used in quantum information processing. We show that when the average variation of the noise is not too large the first order model gives a robust estimate of both the average error affecting the gate set and the gate-dependence of the noise. We also show that the protocol is scalable in the number of qubits comprising the system. The protocol allows the noise to be both time and gate-dependent, and takes into account state preparation and measurement errors.   

Universal Quantum Computation within the Bose-Hubbard Model

We present a novel scheme for universal quantum computation based on spinless bosons hopping on a two-dimensional lattice with on-site interactions. Our setup is comprised of a $2\times n$ lattice for an $n$-qubit system; the two rows correspond to the computational basis states, and a boson in each column encodes a qubit. The system is initialized with $n$ bosons occupying the $n$ sites of the $|0\rangle$ row, and the lattice deep enough to prevent tunneling. Arbitrary single-qubit $X$ rotations are implemented by tuning the tunneling strength between the $|0\rangle$ and $|1\rangle$ sites of the appropriate column, and $Z$ rotations by applying a local potential to the $|1\rangle$ site. Entanglement is generated by hopping between the $|1\rangle$ sites of adjacent qubits; by tuning the on-site interaction strength of the bosons, a non-trivial controlled phase is acquired if these two qubits are in the state $|11\rangle$. Because the quantum information is encoded entirely in the lattice positions of the bosons, the encoded qubits are inherently robust against decoherence. An implementation in terms of ultracold atoms in optical lattices is suggested.   

Optimal Trajectories for Quantum Adiabatic Factoring

We show how a classical multiplication circuit can be expressed as an optimization problem. The circuit can then be effectively run backwards by fixing the output states in the optimization problem and determining the corresponding input states, thereby factoring the output state. This can in turn be expressed as a problem in adiabatic quantum computing. We show by solving a coupled set of Euler-Lagrange equations how (locally) optimal trajectories from initial to final Hamiltonians can be found whose efficacy vastly exceeds that of the usual linear scaling trajectory. Explicit examples will be given for factoring 6-bit integers.   

Deterministic Random-Length Computation with Weakly Entangled Cluster States

Universal quantum computation can be accomplished via single-qubit measurements on a highly entangled resource state, together with classical feedforward of the measurement results. The best-known example of such a resource state is the cluster state, on which judiciously chosen single-qubit measurements can be used to simulate an arbitrary quantum circuit with a number of measurements that is linear in the number of gates. We examine the power of the orbit of the cluster states under GL(2,C), also known as the SLOCC equivalence class of the cluster state, as a resource for deterministic universal computation. We find that, under certain circumstances, these states do indeed constitute resources for such computations, but of random length.   

Logical operator tradeoff for local quantum codes

We study the structure of logical operators in local $D$-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is $d$, then any logical operator can be supported on a set of specified geometry containing $\tilde d$ qubits, where $\tilde d d^{1/(D-1)} = O(n)$ and $n$ is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that two-dimensional codes defined by local commuting projectors admit logical ``string'' operators and are not self correcting.   

Simulating Concordant Computations

A quantum state is called concordant if it has zero quantum discord with respect to any part. By extension, a concordant computation is one such that the state of the computer, at each time step, is concordant. In this talk, I describe a classical algorithm that, given a product state as input, permits the efficient simulation of any concordant quantum computation having a conventional form and composed of gates acting on two or fewer qubits. This shows that such a quantum computation must generate quantum discord if it is to efficiently solve a problem that requires super-polynomial time classically. While I employ the restriction to two-qubit gates sparingly, a crucial component of the simulation algorithm appears not to be extensible to gates acting on higher-dimensional systems.   

Photonic Phase Gate via an Exchange of Fermionic Spin Waves in a Spin Chain

We propose a new protocol for implementing the two-qubit photonic phase gate. In our approach, the $\pi$ phase is acquired by mapping two single photons into atomic excitations with fermionic character and exchanging their positions. The fermionic excitations are realized as spin waves in a spin chain, while photon storage techniques provide the interface between the photons and the spin waves. Possible imperfections and experimental systems suitable for implementing the gate are discussed. [Reference: Phys. Rev. Lett. 105, 060502 (2010)]   

Scalable Neutral Atom Quantum Computer with Interaction on Demand

We propose a scalable neutral atom quantum computer with an on- demand interaction. Artificial lattice of near field optical traps is employed to trap atom qubits. Interactions between atoms can be turned off if the atoms are separated by a high enough potential barrier so that the size of the atomic wave function is much less than the interatomic distance. One-qubit gate operation is implemented by a gate control laser beam which is attached to an individual atom. Two-qubit gate operation between a particular pair of atoms is introduced by leaving these atoms in an optical lattice and making them collide so that a particular two-qubit state acquires a dynamical phase. Our proposal is feasible within existing technology developed in cold atom gas, MEMS, nanolithography, and various areas in optics.   

General-Purpose Quantum Simulation with Prethreshold Superconducting Qubits

We introduce a protocol for the fast simulation of $n$-dimensional quantum systems on $n$-qubit quantum computers with tunable couplings. A mapping is given between the control parameters of the quantum computer and the matrix elements of an $n$- dimensional real (but otherwise arbitrary) Hamiltonian that is simulated in the $n$- dimensional {\it single-excitation subspace} of the quantum simulator. A time- dependent energy/time rescaling minimizes the simulation time on hardware having a fixed coherence time. We demonstrate how three tunably coupled superconducting phase qubits can simulate a three-channel molecular collision using this protocol. The method makes a class of general-purpose time-dependent quantum simulation practical with today's sub-thershold-fidelity qubits.   

Currently Realizable Quantum Error Detection/Correction Algorithms for Superconducting Qubits

We investigate the efficiency of simple quantum error correction/detection codes for zero-temperature energy relaxation. We show that standard repetitive codes are not effective for error correction of energy relaxation, but can be efficiently used for quantum error detection. Moreover, only two qubits are necessary for this purpose, in contrast to the minimum of three qubits needed for conventional error correction. We propose and analyze specific two-qubit algorithms for superconducting phase qubits, which are currently realizable and can demonstrate quantum error detection; each algorithm can also be used for quantum error correction of a specific known error. In particular, we analyze needed requirements on experimental parameters and calculate the expected fidelities for these experimental protocols.   

Interface between Topological and Superconducting Qubits

We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.