Finding Small Two-Qubit Circuits

1
Finding Small Two-Qubit Circuits

• Vivek V. Shende
• Igor L. Markov
• Stephen S. Bullock

2
Outline

• Motivation
• Background
• One Qubit Rotations
• Two Qubit Entanglement
• Our Results

3
We Can Find

• A one-CNOT circuit to prepare a given 2-qubit
  pure state from 0?.
• A two-CNOT circuit to simulate a given 2-qubit
  unitary operator up to relative phase.
• A CNOT-optimal circuit, using at most three
  CNOTs, to simulate a given 2-qubit unitary
  operator.

4
We Can Find

• (If possible) A time, t, for a given Hamiltonian,
  H, such that eiHt, is a CNOT, up to 1-qubit
  gates.
• We can also find the 1-qubit gates.
• An eighteen-CNOT circuit to simulate a given
  three-qubit unitary operator.
• (If a conjecture in Barenco et. Al. 95 holds)

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1-Qubit Rotations
A. Barenco et al., Elementary Gates For Quantum
Computation, PRA 52, 1995.

• We use the following 1-qubit gates
• Any 1-qubit operator can be written as RzRyRz,
  RxRzRx, RxRyRz, etc.
• Three gates needed to simulate a generic operator
  in SU(2), since dim SU(2) 3

6
2-Qubit Entanglement
S. Hill, K. Wooters, Entanglement of a Pair of
Quantum Bits, PRL 78, 1997.

• For a two-qubit state vector, f?, TFAE
• f? is not entangled, ie, f? f1?f0?
• ?0f? ?3f? - ?1f? ?2f? 0
• fTsy?2 f 0
• If u ? U(4) satisfies uTsy?2u sy?2
• (uf)Tsy?2 uf fTuTsy?2 uf fTsy?2f
• Thus u cannot introduce entanglement
• Hence is a wire swap followed by local unitaries
• u (a ? b)(wire swap), then det(a ? b) 1
• If det(u) 1, then u a ? b for a,b ? SU(2)

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The Magic Basis
S. Hill, K. Wooters, Entanglement of a Pair of
Quantum Bits, PRL 78, 1997.

• EET sy?2 iff E-1SU(2)?2E SO(4)
• Proof
• u ? SU(2) iff uTsy?2u sy?2
• Equivalently, (E-1uE)T(E-1uE) 1
• Hence u ? SU(2) iff E-1uE ? SO(4)
• And such E ? U(4) exist

8
The Makhlin Invariants
Yu. Makhlin, Nonlocal Properties of Two-qubit
Gates and Mixed States and Optimization of
Quantum Computations, QIP 1, p. 243, 2002.

• For u,v ? SU(4), sy?2uTsy?2u and sy?2vTsy?2v have
  the same spectra iff there exist a,b,c,d ?SU(2)
  s.t.
• Proof
• In the Magic Basis, this says that UUT is similar
  to VVT iff there exist A,B ? SO(4) such that U
  AVB
• Write UUT A-1VVTA. Since UUT, VVT are
  symmetric, can choose A ? SO(4).
• Then U-1A-1V is in SO(4) and AU(U-1A-1V) V.

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How many gates does it take

• to simulate a generic 2-qubit operator (up to
  global phase)?

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18 Gates Suffice

• Any 2-qubit operator can be simulated by a
  circuit of the form
• Proof Compute invariants

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18 Gates Are Necessary

• Need 15 of the 1-parameter gates since SU(4) is
  15-dimensional
• Need 3 CNOTs to prevent cancellation

a
c
Rx
Rx
Ry
Rz
Ry
Rz
b
d
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How many CNOTs does it take

• to simulate a given
• two-qubit operator?

13
How to count CNOTs

• For u in SU(4), let ?(u) uTsy?2usy?2
• Then u can be simulated using
• 1-qubit gates at most three CNOTs
• 1-qubit gates two CNOTs iff tr?(u) is real
• 1-qubit gates one CNOT iff ?(u)? ?i, ?(u)2 -1
• 1-qubit gates and a wire swap iff ?(u) ?i
• Only 1-qubit gates iff ?(u) ?1
• Proof Compute invariants

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Examples

• Any u in SO(4) can be simulated by two CNOTs and
  some 1-qubit gates
• The wire swap needs three CNOTs
• The two-qubit QFT needs at least three CNOTs and
  three 1-qubit gates

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How many CNOTs does it take

• to simulate a given two-qubit operator up to
  relative phase?

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Simulation up to Relative Phase

• Two CNOTs are sufficient
• Proof for any u ? SU(4), can find a diagonal d
  such that tr?(du) is real.
• Useful because measurement in the computational
  basis kills relative phase

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Can I Simulate the CNOT

• by appropriately timing a given Hamiltonian?

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Simulating CNOT

• Given a Hamiltonian, H, want to determine a time,
  t, for which eiHt differs from CNOT by 1-qubit
  gates
• Compute ?(eiHt) for various values of t

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Application CNOT in the presence of noise

• Fact can time H sx?sx to implement up to CNOT
  1-qubit gates
• Add a noise term, eg., (0.42) I?sz
• Numerical tests yield t0.80587

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Conclusions

• We can find small (often optimal) circuits for
  2-qubit operators

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The End
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Useful fact about sy