Ancilla-driven universal quantum computing

Transcription

1 Ancilla-driven universal quantum computing Vlad Gheorghiu Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. September 30, 2010 Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

2 Outline 1 Introduction to universal quantum computing models 2 The ancilla-driven model 3 Discussions Reference: J. Anders et al, PRA 82, (Rapid Comm.) (2010) A summary of this talk is available online at Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

3 Introduction to universal quantum computing models Introduction What is quantum computation? 1 Preparation: start with n qubits in a fiducial state, e.g. 0 n. 2 Evolve (unitarily) with U SU(2 n ). 3 Measure some of the qubits. Universal quantum computation: the ability of performing any such U + desired measurements! Good enough to be able to perform a small set of gates, that generates SU(2 n ). Arbitrary single qubit gates + CNOT are universal. Solovay-Kitaev theorem: Hadamard H, phase S, π/8 and CNOT can efficiently ɛ-approximate any m-gate unitary (single qubit+cnot ) in Θ(log c (1/ɛ)), c 2. Approximating arbitrary U SU(2 n ) is in general hard! Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

4 Introduction to universal quantum computing models There are 2 well-known universal models of quantum computation. The circuit model: decompose the U using a finite set of universal quantum gates. Figure: Simple quantum circuit. See N&C for comprehensive details. The cluster state model: perform measurements (in arbitrary bases, controlled by the computation) on a highly-entangled state (the cluster state) and effectively simulate the U on a finite set of qubits. Figure: Cluster-state model. Introduced by R. Raussendorf and H.J Briegel, A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001) Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

5 Introduction to universal quantum computing models Both models are universal (and equivalent)! Is there something in between? It is enough to find a scheme for generating arbitrary 1-qubit unitaries + CNOT (or any other entangling gate, like Controlled-phase). Indeed, there is a recent hybrid scheme that does it! Reference: J. Anders et al., Ancilla-driven universal quantum computation, Phys. Rev. A 82, (R) (2010) Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

6 The ancilla-driven model Ancilla-driven model The scheme: Figure: Source: Phys. Rev. A 82, (R) (2010). Illustration of an ancilla-driven computation oon a register consisting of several qubits. A single ancilla, A, is sequentially coupled to one, or at most two, register qubits, R and R, etc., and measured. The coupling, E AR, is fixed throughout the computation while the measurements on the ancilla, indicated by the arrows, can differ. Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

7 The ancilla-driven model Parametrization of single qubit gates Arbitrary single qubit gate U = e iα R n (θ) = e iα [cos ( ) ( ) ] θ θ I sin (n x X + n y Y + n z Z) 2 2 Euler-like representation U = e iα R Z (β)r X (γ)r Z (δ) Yet another representation [V. Danos et al., Phys. Rev. A 72, (2005)] U = e iα J(0)J(β)J(γ)J(δ), J(β) = He i β 2 Z Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

8 The ancilla-driven model Performing arbitrary single qubit gates Single qubit gate scheme: Figure: Source: Phys. Rev. A 82, (R) (2010). Ancilla-driven implementation of a single qubit rotation, J R (β), on a register qubit R. The result of the measurement, j = 0, 1, determines if an X correction appears on the register qubit. The correction can be removed by changing the ancilla measurement bases of future computational operations. The implemented operation on the register qubit is A j J A (β)e AR + A = (X R ) j J R (β), E AR = H A H R CZ AR. Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

9 The ancilla-driven model Performing a two-qubit entangling gate Two-qubit entangling gate scheme: Figure: Source: Phys. Rev. A 82, (R) (2010). Ancilla-driven implementation of a Controlled-Z gate, CZ RR, on two register qubits R and R. The corrections U R (j) and U R (j) are local and can be removed through ancilla-driven single-qubit rotations. A y measurement of the ancilla mediates the entangling operation between R and R, A y j E AR E AR + A = U R (j) U R (j)cz RR, with U R (j) = H R [ (I + izr )/ 2 ] (Z R ) j. Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

10 Discussions Universal entangling operations Ancilla-driven quantum computation with a fixed interaction E AR = H A H R CZ AR allows universal quantum computation. In MBQC only standard X, Y and Z corrections occur. Here the errors are more general, but still localized. Can one use some other class of entangling operations E AR? Key result: only 2 classes of couplings are universal. They must be locally equivalent to either CZ gate or CZ + SWAP gate Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11

11 Discussions Experimental realizations Authors claim: suited for many physical realizations. 1 Trapped atoms in an optical lattice addressed by ancilla-marker atoms, which interact via cold collisions to generate CZ gates 2 Ions in microtraps and ancilla read-write ion that interacts by laser-induced state-dependent pushing forces. Using optimized control pulses it may be possible to generate the E AR operation efficiently and robustly in a single step. 3 See the paper for more proposals... Take-home message: system + ancilla + fixed 2-qubit interaction (E AR ) + arbitrary single-qubit measurements. Probably generalizable to qudits! (my guess, although I have not really checked). Vlad Gheorghiu (CMU) Ancilla-driven universal quantum computing September 30, / 11