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Efficient simulation of quantum computers: the computers vs classical computers Classical computers

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  • Efficient simulation of quantum computers: theGottesman-Knill theorem or an application of group

    theory to quantum information

    Vlad Gheorghiu

    Department of PhysicsCarnegie Mellon University

    Pittsburgh, PA 15213, U.S.A.

    January 28, 2008

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 1 / 17

  • Outline

    1 Quantum computers vs classical computersClassical computersQuantum computers

    2 Simulating quantum computers: main issues

    3 The Pauli groupThe Pauli group on one qubitThe Pauli group on n qubits

    4 The Clifford groupThe Clifford group on one qubitThe Clifford group on n qubits

    5 Stabilizer groups

    6 The Gottesman-Knill theorem

    7 References

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 2 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}

    The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}

    The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Classical computers

    A classical computer is a machine that manipulates classical data (orbits) according to a list of instructions

    Its main ingredients are bits, wires and gates

    More formally, a classical computer takes an n-bit input{a1, a2, . . . , an} and computes a functionf (a1, a2, . . . , an) = {b1, b2, . . . , bm}The result of the computation is the m-bit string {b1, b2, . . . , bm}The function f depends on the application

    The function can be implemented by a set classical gates, like AND,NOT, OR, XOR etc

    The NAND gate is universal for classical computing, that is, anyother gate can be simulated using the NAND gate alone

    Some functions can be very hard to compute, and best knownalgorithms require a computation time that scales exponentially withthe number of input bits. Examples: factoring, traveling salesmanproblem etc.

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 3 / 17

  • Quantum computers vs classical computers Quantum computers

    A quantum computer is a device for computation that makes directuse of quantum mechanical phenomena, such as superposition andentanglement, to perform quantum operations on quantum data, orqubits (Wikipedia)

    Figure: A quantum computer

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 4 / 17

  • Quantum computers vs classical computers Quantum computers

    A quantum computer is a device for computation that makes directuse of quantum mechanical phenomena, such as superposition andentanglement, to perform quantum operations on quantum data, orqubits (Wikipedia)

    Figure: A quantum computer

    Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 28, 2008 4 / 17

  • Quantum computers vs classical computers Quantum computers

    T