Schrödinger’s Elephants & Quantum Slide Rules A.M. Zagoskin (FRS RIKEN & UBC) S. Savel’ev (FRS RIKEN & Loughborough U.) F. Nori (FRS RIKEN & U. of Michigan) Solving NP-complete problems with approximate adiabatic evolution

Standard quantum computation Consecutive application of unitary transformations (quantum gates) Consecutive application of unitary transformations (quantum gates) Problem encoded in the initial state of the system Problem encoded in the initial state of the system Solution encoded in the final state of the system Solution encoded in the final state of the system  digital operation Examples: Shor’s algorithm Grover’s algorithm quantum Fourier transform quantum Fourier transform Precise time-domain manipulations Precise time-domain manipulations  complex design and extra sources of decoherences Problem and solution encoded in fragile strongly entangled states of the system Problem and solution encoded in fragile strongly entangled states of the system  effective decoherence time must be large Quantum error-correction (to extend the coherence time of the system) Quantum error-correction (to extend the coherence time of the system)  overhead (threshold theorems: (!)) Aharonov, Kitaev & Preskill, quant-ph/

Adiabatic quantum computation Continuous adiabatic evolution of the system Continuous adiabatic evolution of the system Problem encoded in the Hamiltonian of the system Problem encoded in the Hamiltonian of the system Solution encoded in the final ground state of the system Solution encoded in the final ground state of the system Farhi et al., quant- ph/ ; Science 292(2001)472 The approach is equivalent to the standard quantum computing The approach is equivalent to the standard quantum computing Aharonov et al., quant- ph/ “Space-time swap”: the time- domain structure of the algorithm is translated to the time- independent structural properties of the system “Space-time swap”: the time- domain structure of the algorithm is translated to the time- independent structural properties of the system Ground state is relatively robust Ground state is relatively robust  much easier conditions on the system and its evolution Well suited for the realization by superconducting quantum circuits Well suited for the realization by superconducting quantum circuits Kaminsky, Lloyd & Orlando, quant-ph/ Grajcar, Izmalkov & Il’ichev, PRB 71(2005)144501

Travelling salesman’s problem* N points with distances d ij N points with distances d ij Let n ia =1 if i is stop #a and 0 otherwise; there are N 2 variables n ia (i,a = 1,…,N) Let n ia =1 if i is stop #a and 0 otherwise; there are N 2 variables n ia (i,a = 1,…,N) The total length of the tour The total length of the tour *See e.g. M. Kastner, Proc. IEEE 93 (2005) 1765

Travelling salesman’s problem The cost function The cost function

Travelling salesman’s problem Ising Hamiltonian Ising Hamiltonian

Spin Hamiltonian Adiabaticity parameter Adiabaticity parameter

Adiabatic optimization

Approximate adiabatic optimization vs. simulated annealing

RMT theory near centre of spectrum* RMT theory near centre of spectrum* Diffusive behaviour Residual energy β = 1 (GOE); 2 (GUE) Simulated annealing** Simulated annealing** ζ ≤ 6 *M. Wilkinson, PRA 41 (1990) 4645 **G.E. Santoro et al., Science 295 (2002) 2427

Running time vs. residual energy Classical/quantum simulated annealing (classical computer) Classical/quantum simulated annealing (classical computer) Approximate adiabatic algorithm (quantum computer) Approximate adiabatic algorithm (quantum computer)

Solution is encoded in the final ground state Solution is encoded in the final ground state Error produces unusable results (excited state does not, generally, encode an approximate solution) Error produces unusable results (excited state does not, generally, encode an approximate solution) Objective: minimize the probability of leaving the ground state Objective: minimize the probability of leaving the ground state Solution is a (smooth enough) function of the energy of the final ground state Solution is a (smooth enough) function of the energy of the final ground state Error produces an approximate solution (energy of the excited state is close to the ground state energy) Error produces an approximate solution (energy of the excited state is close to the ground state energy) Objective: minimize the average drift from the ground state Objective: minimize the average drift from the ground state Relevant problems: Relevant problems: finding the ground state energy of a spin glass traveling salesman problem AQC vs. Approximate AQC

Generic description of level evolution: Pechukas gas* *P. Pechukas, PRL 51 (1983) 943

Pechukas gas kinetics

Pechukas gas kinetics: taking into account Landau-Zener transitions

Pechukas gas flow simulation

Level collisions and LZ transitions

“Diffusion” from the initial state

Analog vs. digital

4-flux qubit register *M. Grajcar et al., PRL 96 (2006)

Conclusions Eigenvalues behaviour is not described by simple diffusion Eigenvalues behaviour is not described by simple diffusion Marginal states behaviour qualitatively different: adiabatic evolution generally robust Marginal states behaviour qualitatively different: adiabatic evolution generally robust Analog operation of quantum adiabatic computer provides exponential speedup Analog operation of quantum adiabatic computer provides exponential speedup Advantages of Pechukas mapping: exact, provides intuitively clear description and controllable approximations (BBGKY chain) Advantages of Pechukas mapping: exact, provides intuitively clear description and controllable approximations (BBGKY chain) In future: external noise sources; mean-field theory; quantitative theory of a specific algorithm realization; investigation of the class of AA-tractable problems In future: external noise sources; mean-field theory; quantitative theory of a specific algorithm realization; investigation of the class of AA-tractable problems