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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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}

11. 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?)

12. 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?