An Arbitrary Two-qubit Computation in 23 Elementary Gates or Less

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An Arbitrary Two-qubit Computationin 23
Elementary Gates or Less

• Stephen S. Bullock
• and Igor L. Markov

University of Michigan Departments of
Mathematics and EECS
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Outline

• Introduction
• A crash-course in quantum circuits
• Prior work
• Synthesis by matrix factorizations
• Our contribution
• Entanglers and disentanglers
• Recognizing tensor products
• Circuit decompositions
• Conclusions and on-going work

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Introduction

• Abstract synthesis of logic circuits
• Input a function that represents a computation
• Output a circuit that implements that function
• Minimize circuit size
• Our focus two-qubit quantum computation
• Quantum states are complex vectors
• Computations and gates are 4x4-unitary matrices
• Existing commercial applications
• Secure communication quantum key
  distribution(circuits are small, but gates are
  very expensive)
• Future applications quantum computing

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Quantum Information
Special session tomorrow Cambridge, NIST Los
Alamos, and Michigan

• Most popular carriers
• Polarization of an individual photon
• Spin of an individual nucleus or electron
• Energy-level of an individual electron
• Common features
• Basis states, e.g., ? and ? or ? and ?
• Linear combinations are allowed
• ?0gt?1gt ?2?21 ?,? are complex
• Fundamental change from classical info

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Classical Models Quantum Models

• 0-1 strings
• E.g., one bit0,1
• Bool. Functions
• Gates circuits
• Primary outputs

• Lin. combinationsof 0-1 strings
• E.g., one qubit?0gt?1gt
• Matrices
• Gates circuits
• Probabilisticmeasurement
• Mostly ignored here

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From Classical to Quantum

• All quantum computations M must be unitary
• M x M-conjugate-transpose I
• ? M-1 (recall reversible computations)
• A conventional reversible gate/computationcan be
  extended by linearity
• E.g., a quantum inverter swaps 0gt and 1gt
• Maps the state (0gt1gt)/?2 to itself
• Can apply an inverter on one of two qubits
• E.g., (00gti11gt)/?2 ? (01gti10gt)/?2

0 1 1 0
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Quantum Circuits

• Can apply an inverter on one of two qubits
• E.g., (00gti11gt)/?2 ? (01gti10gt)/?2
• How do we describe this computation?
• Tensor product Identity?NOT
• More generally A?B

0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
CNOT gate
x
x
y?x
y
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Elementary Gates Q. Computation
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Technology-indep. Synthesis

• Input Unitary 4x4-matrix M
• Generic quantum computation on 2 qubits
• Output circuit in terms of elem. gates that
  implements M up to a constant
• Minimize circuit cost
• E.g., gate count or ? (gate costs)
• Solutions existiff the gate library is universal

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Phase can be ignored
Gate 2
Gate 3
Gate 1
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Previous Work

• Proof of universality is constructive
  Barenco et. al 95 in Phys. Rev. A
• Can be interpreted as a synthesis algorithm
• However, no attempt to minimize gates
• Can be viewed as matrix factorization
• Cybenko 01
• MQR with unitary Q upper-triangular R
  (M unitary ? R diagonal)
• We count gates, and the answer is

61
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Our Work (1)

• Two new synthesis procedures (2 qubits)
• Based on a different matrix factorization(related
  to SVD and KAK decompositions)
• Also reduce generic synthesis to diag synth
• Small circuits for diagonal computations
• Smaller overall circuits than QR (Cybenko)
• Constructive worst-case bounds
• Any circuit in gates or less
• Any circuit with 15 non-constant gates

23
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Our Work (2)

• Lower bounds
• ? two-qubit computations (most of them)that
  require at least 17 elementary gates
• At least 15 non-const gates
• At least 2 CNOTs
• Bounds are not constructive and not tight,except
  for 15 non-const gates
• We never use temporary storage qubitsbut this
  could lead to smaller gate counts

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The Entangler and Disentangler

• Computational basis
• 00gt,01gt,10gt and 11gt
• The entangler computation maps00gt to
  (00gt11gt)/?2, etc.
• The disentangleris E-1E
• Key lemma
• If UA?B, then EUE has only real entries
• An efficient way to recognize tensor products

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Circuits For E and E

• A specific circuit for the entangler E
• Sdiag(1,i) counts as one elementary gate
• The Hadamard gate H counts as two
• E is implemented by reversing the diagram
• Change S to S-1diag(1,-i)

7 elem. gates
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Our (Key) Synthesis Procedure

• The canonical decomposition for 2-qubit
  computations
• ? U ? K1,K2 and ? such that UK1?K2
• E?E is diagonal (5 gates)
• K1,K2 have only real entries
• The terms K1, K2 and ? can be found explicitly
• Numerical analysis polar and spectral
  decompositions
• Reduce K1 and K2 to tensor products using
  entanglers
• EUEE(A?B)EE?EE(C?D)E
• A,B,C and D are one-qubit computations 3 gates
  each
• Note that E and E are the same for any input

Rz
Rz
Rz
Rz
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Details (1)

• After the initial divide-and-conquermany gate
  cancellations can be made
• This brings down max gates to 28
• Only 15 of them depend on input,which matches an
  a priori lower bound
• Further reductions from the analysisof E(A?B)E
  and E(C?D)E
• Max gates reduced to
• However, 19 gates depend on the input

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Details (2)

• The structure of the 23-gate circuit
• For additional details, see
• Our paper in Physical Review A
• http//xxx.lanl.gov/abs/quant-ph/0211002

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Validation of Our Synthesis Algo

• Implementation in C
• Produces 23 gates for randomly- generated
  4x4-unitary matrices
• Can capture structureseveral examples in
  quant-ph/0211002
• Optimal results for any A?B circuit(QR
  decomposition ? typically 61 gates)
• For 2-qubit Fourier transform a circuit with
  minimal of CNOT gates

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Conclusions and On-going Work

• First generic synthesis algorithm to capture
  circuit structure, e.g., A?B
• On-going work
• Lower and upper bounds of gates (almost
  done)
• Solved synthesis of n-qubit diagonal
  computations (produce circuits within 2x from
  optimal)

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Thank you!

• Questions are welcome

Ion Traps
Nuclear Magnetic Resonance
Quantum dots
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Thank you!