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Translating Quantum Gate  Adiabatic

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. Remember XOR?.

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Translating Quantum Gate  Adiabatic

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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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