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Adiabatic Quantum Computing: Translating Gates to Adiabatic Quantum Gate

Introduction to translating Boolean logic to adiabatic gates, focusing on building blocks and techniques for reversible quantum computing. Projects explore gate optimization and modularization for efficient circuit design.

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Adiabatic Quantum Computing: Translating Gates to Adiabatic Quantum Gate

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

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

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

  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 applyingoutputs 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=xtif 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 asnew 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?

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