Efficient measure of scalability

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

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

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

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

5. Type of errors Type of errors: how constant is E? Type of errors: how are the non-null coefficients in H ?  Uniform:All the qubits perceive the same error: j= , j,k=   Gaussian: The qubits react independently: the j, j,kare drawn from a Gaussian distribution with center ,  and dispersion ,  respectively.

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

7. General results General results  The decay is essentially exponential: Numerically:

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

9. The initial decay rate  General results

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

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

12. Conclusions 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. (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.

13. References Questions? On the fidelity as a useful tool: J. Emerson et al., quant-ph/0503243 (2005) C. A. Ryan et al., quant-ph/0506085 (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) J. Emerson et al., PRL 89, 284102 (2002) D. Poulin et al., PRA 68, 022302 (2003)