Translating Quantum Gate  Adiabatic Based on Gates for Adiabatic Quantum Computing by Richard H. Warren, arXiv:1405.2354, 2014

Remember XOR? Introduce an ancilla qubit to make it, one solution: A+B+Y+4a+2AB-2AY-4Aa-2BY-4Ba+4Ya 1 1 1

Remember XOR? Introduce an ancilla qubit to make it, one solution: A+B+Y+4a+2AB-2AY-4Aa-2BY-4Ba+4Ya Verified: a1 a2 a3 a4 b12 b13 b14 b23 b24 b34 1 4 2 -2 -4   q1 q2 q3 q4 Objective 9

From Boolean Logic to Gates Boolean gates are already revisible But gates do not fit 2-local Ising model – or so it seems at first Problem: need steps to iterate through sequence of gates, options: Need a counter, +1 in each step adiabatic gate only “fires” when counter=i for ith gate use “outputs” of Hamiltonian Hi to drive “inputs” of Hi+1 we’ve done this before: compose adder  multi-bit adder

From Boolean Logic to Gates (2) Problem: need building blocks for gates CNOT same truth table as XOR just copy in-x  out-x Reversible by applying outputs as new inputs All other gates are One-bit gates + CNOT

From Boolean Logic to Gates (3) if in_c1 & in_c2 then result = not(target) else result = target Toffoli gate: 6 CNOT + 1-bit gates Adiabatic Toffoli: 1 CNOT, 6 qubits: controls c1,c2; target t, result r, ancillas a,b xb=xc1xc2 xb =1 iff xc1=1=xc2 if xb=0xr=xt if xb=1xr=1-xt

From Boolean Logic to Gates (4) Adiabatic Toffoli: Same as CNOT  XOR Qubo: 2xbxt-2(xb+xt)xr-4(xb+xt)xa+4xrxa+xb+xt+4xa xb=xc1xc2 Qubo: xc1xc1-2(xc1+xc2)xb+3xb Add both Qubos: -4xaxb+4xaxr-4xaxt-2xbxc1-2xbxc2-2xbxr+2xbxt+xc1xc1-2xrxt+4xa+4xb+xr+xt Hamiltonian coefficients  Reversible by applying outputs as new inputs 6 qubits, XOR, equal inputs outputs

From Boolean Logic to Gates (5) Fredkin Gate: if c then swap i,j, or: m=(1-c)i+cj, p=ci+(1-c)j Adiabatic Fredkin Qubos: -ci+cj+2cim-2cjm+i-2im+m ci-cj-2cip+2cjp+j-2jp+p Not in 2-local Ising formatancillas, a=cm, b=cp Add equal qubos: cm-2(c+m)a+3a and cp-2(c+p)b+3b -4ac+2ai-2aj-4am+4bc-2bi+2bj=4bp+2cm+2cp-2im-2jp+6a+6b+i+j+m+p Reversible (outputs m,p  new inputs i,j) 7 qubits, swap+2xequal inputs outputs

From Boolean Logic to Gates (6) Hadamard Gate: Let |0> and |1> be 1st/2nd vectors in basis  for 2-dimensional space H maps |0>  (|0>+|1>)/2 and |1>(|0>-|1>)/2 Since =a|0>+b|1>, H=aH|0>+bH|1>=((a+b)|0>+(a-b)|1>)/2 (Fourier) Matrix notation: a2+b2=1  Vector notation: H(a,b)=( (a+b)/2, (a-b)/2 ) Adiabatic Hadamard: for qubits i,j with local field hii and hjj (weights) H(i,j)( (hi+hj)/2, (hi -hj)/2) so for output qubits q,p, their weights hq=(hi+hj)/2, hp=(hi-hj)/2 4 qubits

From Boolean Logic to Gates (7) Adiabatic Hadamard: for qubits i,j with local field hii and hjj (weights) H(i,j)( (hi+hj)/2, (hi-hj)/2) so for output qubits q,p, their weights hq=(hi+hj)/2, hp=(hi-hj)/2 Reversible since hi=(hp+hq)/2 and hj=(hp-hq)/2 Assumes hi2+ hj2=1 So if (hi,hj)=(1,0) and (hp,hq)=(0,1)  hp2+hq2=1 This is were it gets wild, claim: Need i=1 s.t. hii reflects correct local field (weights), 2 options: Coupler Ji,k between i and k needs to be adjusted (“balanced”) Add penalty –xi , where xi є{0,1}

Implications (1) Can auto-translate simple gates  adiabatic Open problems: Other simple qubit Pauli gates Harder problems: Generalization to any spins  infeasible or just more qubits? Project 1: create translator gates  adiabatic Try for sample circuits Adder Toffoli composed of simple gates Etc. Project 2: optimize circuits Try to replace know sub-circuits with cheaper ones E.g., multi-gate Toffoli  adiabatic Toffoli (feasible?)

Implications (2) Can we auto-translate adiabatic  simple gates ??? Project 3: start with boolean logic qubos  should be feasible Could also start w/ S. Pakin’s quasm macros Limited to netlist logic macros that are required Generalize to other qubos Need to modularize, find building blocks  harder Try for sample circuits Question: Can we define the subset of programs that can be translated from AB and BA?