Efficient measure of scalability Cecilia López, Benjamin Lévi, Joseph Emerson, David Cory Department of Nuclear Science & Engineering, Massachusetts Institute of Technology ( through fidelity decay )

Definitions Identifying errors through fidelity decay Target Control of the system We must fight against errors. We need to identify errors.   Quantum process tomography Inefficient!  Other proposals: less information but at a lower cost  Fidelity decay   

is a random rotation that spans U(2): Definitions Using randomness to explore the Hilbert space with , ,  drawn randomly. We use a random operator as the evolution operator U :   

is a random rotation that spans U(2): Definitions Using randomness to explore the Hilbert space with , ,  drawn randomly. E is the error arising from an imperfect implementation of the Identity operator: with  j,  j,k small. We use a random operator as the evolution operator U :    (an ensemble of realizations)

Type of errors Type of errors: how constant is E ? Coherent: The parameters  j,  j,k remain constant. Incoherent: The parameters  j,  j,k change after certain time – the correlation time. Long correlation time  order of the experiment length Short correlation time  order of the implementation of a gate length  Uniform: All the qubits perceive the same error:  j = ,  j,k =   Gaussian: The qubits react independently: the  j,  j,k are drawn from a Gaussian distribution with center ,  and dispersion  ,   respectively. Type of errors: how are the non-null coefficients in H  ?

General results  The decay is essentially exponential: Numerically: General results We can fit  At long times, the state is completely randomized:   

General results  The decay is essentially exponential: Numerically: General results

 The decay is essentially exponential: Numerically: General results We can fit  At long times, the state is completely randomized:    Analytically: Confirmed by expressions for H  with one-qubit terms only.

General results The initial decay rate 

Promising! Inefficient! Hard to engineer! The initial decay rate    Locality of errors

For instance: Advantages:  Initial state preparation is less critical  Less measurements

General results  The decay is essentially exponential  The fidelity decay rate is related to type and strength of the noise  The initial decay rate  is independent of the type of errors   can be used to address the question of the locality of errors  The locality of errors is key to determine whether we need non-local gates to correct them: the need of non-local gates would imply the lack of scalability of that particular system. Conclusions (analytically for one-qubit terms, numerically including two-qubit terms)  We are working on the experimental implementation of this scheme in liquid NMR, with a 4-qubit molecule.

References Questions? J. Emerson et al., PRL 89, (2002) D. Poulin et al., PRA 68, (2003) On the fidelity as a useful tool: J. Emerson et al., quant-ph/ (2005) C. A. Ryan et al., quant-ph/ (2005) On the mathematical background for our calculations: P. W. Brouwer and C. W. J. Beenakker, J. Math. Phys. 37, 4904 (1996) P. A. Mello, J. Phys. A 23, 4061 (1990) S. Samuel, J. Math. Phys. 21, 2695 (1980)