Quantum hardware characterized by programmable bose-hubbard hamiltonians

Abstract

An apparatus includes a first group of superconducting cavities and a second group of superconducting cavities, each of which is configured to receive multiple photons. The apparatus includes couplers, where each coupler couples one superconducting cavity from the first group with one cavity from the second group such that the photons in the coupled superconducting cavities interact. A first superconducting cavity of the first group is connected to a second superconducting cavity of the second group, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities. The first superconducting cavity is coupled to at least one other superconducting cavity of the first group to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to at least one other superconducting cavity of the second group to which the first superconducting cavities are coupled.

What is claimed is: 
     
         1 . An apparatus comprising:
 a first plurality of superconducting cavities each configured to receive a plurality of photons;   a second plurality of superconducting cavities each configured to receive a plurality of photons; and   a plurality of couplers, wherein each coupler couples one superconducting cavity from the first plurality of superconducting cavities with one superconducting cavity from the second plurality of superconducting cavities such that the photons in the coupled superconducting cavities interact; and   wherein a first superconducting cavity of the first plurality of superconducting cavities is connected to a second superconducting cavity of the second plurality of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first plurality of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one or more of the other superconducting cavities of the second plurality of superconducting cavities to which the first superconducting cavities are coupled.   
     
     
         2 . The apparatus of  claim 1 , wherein each coupler is configured to annihilate a photon in one superconducting cavity and create a photon in a different superconducting cavity. 
     
     
         3 . The apparatus of  claim 1 , wherein at least one of the couplers comprises a Josephson Junction. 
     
     
         4 . The apparatus of  claim 1 , wherein a Hamiltonian characterizing the apparatus is:
   Σ i   h   i   n   i +Σ i,j   t   ij (α i   † α j +h.c.)+Σ i   U   i   n   i ( n   i −1),
   where n i  is a particle number operator and denotes occupation number of a cavity mode i, α i   †  is a creation operator that creates a photon in cavity mode i, α j  is an annihilation operator that annihilates a photon in cavity mode j, h i  corresponds to a site disorder, U i  corresponds to an on-site interaction, t i,j  are the hopping matrix elements, and h.c. is hermitian conjugate.   
     
     
         5 . The apparatus of  claim 4 , wherein the plurality of couplers is trained to produce an output desired probability density function at a subsystem of interest at an equilibrium state of the apparatus. 
     
     
         6 . The apparatus of  claim 4 , wherein the apparatus is trained as a Quantum Boltzmann Machine. 
     
     
         7 . The apparatus of  claim 1 , wherein a Hamiltonian characterizing the apparatus is: 
       
         
           
             
               
                 
                   
                     ∑ 
                     i 
                   
                    
                   
                       
                   
                    
                   
                     
                       h 
                       i 
                     
                      
                     
                       n 
                       i 
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       t 
                       ij 
                     
                      
                     
                       ( 
                       
                         
                           
                             a 
                             i 
                             † 
                           
                            
                           
                             a 
                             j 
                           
                         
                         + 
                         
                           h 
                           . 
                           c 
                           . 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     i 
                   
                    
                   
                       
                   
                    
                   
                     
                       U 
                       i 
                     
                      
                     
                       
                         n 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             n 
                             i 
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       U 
                       ij 
                     
                      
                     
                       n 
                       i 
                     
                      
                     
                       n 
                       j 
                     
                   
                 
               
               , 
             
           
         
         wherein n i  is a particle number operator and denotes occupation number of a cavity mode i, α i   †  is a creation operator that creates a photon in cavity mode i, α i  is an annihilation operator that annihilates a photon in cavity mode j, h i  corresponds to a site disorder, U i  corresponds to an on-site interaction, t i,j  are the hopping matrix elements, and h.c. is hermitian conjugate. 
       
     
     
         8 . The apparatus of  claim 7 , wherein the apparatus is operable to evolve adiabatically to a ground state of a problem Hamiltonian H p =Σ i h i n i +Σ i U i n i (n i −1)+Σ i,j U ij n i n j . 
     
     
         9 . The apparatus of  claim 7 , wherein the apparatus is operable to evolve adiabatically from a Mott-insulator state to a superfluid state, and wherein an initial Hamiltonian of the apparatus is H i =Σ i,j t ij (α i   † α j +h.c.). 
     
     
         10 . The apparatus of  claim 7 , wherein the apparatus is operable to evolve adiabatically from a Mott-insulator state to a ground state of a problem Hamiltonian H p =Σ i h i n i +Σ i U i n i (n i −1)+Σ i,j U ij n i n j , and wherein an initial Hamiltonian of the apparatus is H i =Σ i,j t ij ; (α i   † α j +h.c. 
     
     
         11 . The apparatus of  claim 1 , wherein the apparatus is configured to respond to an external field ε(t) and a Hamiltonian characterizing the apparatus in the external field is:
   Σ i   h   i   n   i +Σ i,j   t   ij (α i   † α j +h.c.)+Σ i   U   i   n   i ( n   i −1)+Σ i [ε( t )α i   † +ε( t )*α i   ]+H   SB ,
 
   wherein 
     H   SB =Σ i Σ υ [κ i,υ (α i   b   υ   † +α i   †   b   υ )+λ i,υ α i   † α i ( b   υ   +b   υ   † )], and
 
 wherein n i  is a particle number operator, ε(t) is a slowly-varying envelope of an externally applied field to compensate for photon loss, H SB  is a Hamiltonian of the interaction between the apparatus and a background bath in which the apparatus is located, b υ , and b υ   †  are annihilation and creation operators for a bosonic background bath environment, κ i,υ  is a strength of apparatus-bath interactions corresponding to exchange of energy, h i  corresponds to a site disorder, U i  corresponds to an on-site interaction, t i,j  are the hopping matrix elements, and λ i,υ  corresponds to a strength of local photon occupation fluctuations due to exchange of phase with the bath. 
 
     
     
         12 . The apparatus of  claim 11 , wherein the apparatus is operable to be dissipatively-driven to a ground state of a problem Hamiltonian. 
     
     
         13 . The apparatus of  claim 1 , wherein at least one cavity is a 2D cavity. 
     
     
         14 . The apparatus of  claim 1 , wherein each cavity is a 3D cavity. 
     
     
         15 . The apparatus of  claim 1 , wherein each superconducting cavity in the first plurality of superconducting cavities is connected to a superconducting cavity in the second plurality of superconducting cavities. 
     
     
         16 . A method comprising:
 providing the apparatus of  claim 1  in an initial Mott-insulated state;   causing a quantum phase transition of the apparatus from the initial Mott-insulator state to a superfluid sate; and   adiabatically guiding the apparatus to a problem Hamiltonian.   
     
     
         17 . The method of  claim 16 , further comprising:
 causing a quantum phase transition of the apparatus from the superfluid state to a final Mott-insulator state; and   reading the state of each superconducting cavity in the apparatus.