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Quantum Circuit Decomposition from unitary matrices into elementary gates

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Prologue In classical logic synthesis, one may trivially decompose any boolean function into an OR of ANDs (sum of products) Local optimizations may then be applied to shrink the resulting circuit Can the same be done in the quantum case?

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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References The A. Barenko et. Al. paper, and how to write a controlled-U gate in elementary gates – U(2) and SU(2) matrices – Controlled-U gates The Cybenko paper, and how to write an arbitrary unitary matrix in elementary gates – QR decomposition – Making it a circuit

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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The controlled-U The block-matrix form of a controlled-U gate These can be decomposed into – CNOT gates – 1-qubit rotations

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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One Qubit Rotations Let U be a SU(2) matrix. U must take the form Where

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One Qubit Rotations Define So that

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Some Quick Facts R takes sums to products (R=R z or R y ) R(0)=I. So: Finally,

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Circuit Decompositions The A. Barenko et. Al. paper, and how to write a controlled-U gate in elementary gates – U(2) and SU(2) matrices – Controlled-U gates The Cybenko paper, and how to write an arbitrary unitary matrix in elementary gates – QR decomposition – Making it a circuit

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Controlled-U Gates Consider the controlled-U gate Claim: this circuit is equivalent UBAC

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Controlled-U Gates Check this circuit on basis states One observes BAC

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Controlled-U Gates Check this circuit on basis states One observes BAC

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Controlled-U Gates Check this circuit on basis states One observes BAC

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Controlled-U Gates Check this circuit on basis states One observes BAC

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Controlled-U Gates Check this circuit on basis states One observes And similarly, BAC

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Controlled-U Gates Check this circuit on basis states One observes And similarly, BAC

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Controlled-U Gates Check this circuit on basis states One observes And similarly, BAC

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Controlled-U Gates Check this circuit on basis states One observes And similarly, BAC

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Controlled-U Gates Check this circuit on basis states One observes And similarly, BAC

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Controlled-U Gates Check this circuit on basis states By linearity, this circuit performs controlled-U BAC

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Controlled-U Gates If U is in U(2) (as opposed to SU(2)), – write U=d U, where d 2 =det U, U in SU(2) Then U U D == BAC D

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Higher Order Controlled-U Gates Recall (from two weeks ago) – Where V is a square root of U. This generalizes straight-forwardly to higher numbers of qubits U = VV*V

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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QR-Decomposition Given a vector (a,b), this SU(2) matrix kills the second coordinate

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QR-Decomposition The vector (a,b) might be sitting inside a matrix: Think of this as a rotation of the plane in which the 3 rd and 4 th coordinates live Note that this matrix is unitary

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Making it a Circuit The matrix used to kill coordinates in the bottom row looks like This is a (higher order) controlled-U gate!

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QR-Decomposition One may iterate this process

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QR-Decomposition One may iterate this process

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QR-Decomposition One may iterate this process

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QR-Decomposition One may iterate this process

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QR-Decomposition One may iterate this process

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QR-Decomposition One may iterate this process

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QR-Decomposition This yields the formula – Where X was the original matrix, the U i are planar rotations, and R is upper triangular with nonnegative real entries on the diagonal

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QR-Decomposition Inverting the Q,

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QR-Decomposition If X is unitary, then R is the product of unitary matrices and hence unitary. A triangular unitary matrix must be diagonal A diagonal unitary matrix with nonnegative real entries must be the identity

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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Making it a Circuit The matrix used to kill coordinates in the bottom row looks like This is a (higher order) controlled-U gate!

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Making it a Circuit Need to make other planar rotations controlled-U gates For some j, given an operator P j P j UP j -1 is a rotation in the j,j+1 plane. (where U is a rotation in the n-2,n-1 plane)

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Making it a Circuit Built the operator out of NOT and CNOT gates How to do it for the case of 4 qubits, j=5

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Making it a Circuit Built the operator out of NOT and CNOT gates How to do it for the case of 4 qubits, j=5 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1

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Making it a Circuit Built the operator out of NOT and CNOT gates How to do it for the case of 4 qubits, j=5 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1

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Making it a Circuit The general case is not much harder – First, flip all bits that are 0 in both j,j+1 – Then, CNOT every remaining bit that is zero in j+1, controlling by the unique bit that is 1 in j+1 and 0 in j – Finally, switch this unique bit with the low bit

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Objectives Introduce the controlled-U gate – Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates Introduce the QR-decomposition Use QR to decompose a unitary matrix into controlled-U gates – Conclude that any operator can be built of CNOT gates and 1-qubit rotations

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Conclusion A unitary matrix can be written as a product of planar rotations A planar rotation can be written as ZUZ -1, where Z can be decomposed into CNOT and NOT gates, and U is a (higher order) controlled-U gate A higher order controlled-U gate can be written as a sequence of CNOT gates and singly controlled-U gates A controlled-U gate can be written as a sequence of CNOT gates and one-qubit rotations

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Epilogue The number of gates in this decomposition is exponential in the number of qubits For certain operators, much smaller circuits are known to exist Can we automate the process of moving towards these?

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Reduction Could try to shrink a long circuit by local optimization techniques One experimentally observed obstacle: long chains of CNOT gates These long chains of CNOTs result from certain identities

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Reduction Could apply classical techniques…

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