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UMD PHYS 402 - Quantum Computation and the Bloch Sphere

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Quantum Computationand the Bloch SphereFred WellstoodJoint Quantum InstituteandCenter for Superconductivity Research Department of PhysicsUniversity of Maryland, College Park, MD(March 24, 2008)In principle, a computer can be built that uses quantum mechanics to perform useful calculations.A quantum computer would be built from quantum bits or "qubits",individual quantum system with two basis states, |0> and |1>The qubits are coupled together and logic operations are performed by manipulating the quantum state of the entire system. Example: NOT on single qubit:|0> |1> |1> |0> α|0> +β|1> α|1> +β|0>Example: Phase gate on single qubit:|0> |0> |1> eiφ|1> α|0> +β|1> α|1> +eiφβ|0>Quantum Mechanics and Quantum Computingoperations need to work on superposition states!The Principle of SuperpositionSuppose |0> and |1> are two allowed quantum states of a system, then the system can exist in any linear superposition of these stateswhere αand βare complex numbers>+>= 1|0|βαψBut we don’t see such states in everyday objects "Schrodinger's cat paradox" (Schrodinger, 1935) if true in macroscopic objects +livedead"macroscopic quantum superposition"Quantum Mechanics and Quantum Computing>+>=1|0|βαψSuperposition State- state must be normalized to unity122=+βα- probability amplitudes α and β can be complex numbers- then define )cos(θα=)sin(θβφie=1)(sin)(cos)sin()cos(222222=+=+=+θθθθβαφie- an overall phase factor has no effect, so we can choose α to be real>+>=>+>= 1|)sin(0|)cos(1|0|θθβαψφie- so we can always write a superposition state in the form:|0>|1>zyxθφSuperposition States as Points on the Bloch Spheresphere with radius R=1 …..this is the “Bloch Sphere”>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψ 1|2sin0|2cosθθφie|0>zyx>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψ0|1|20sin0|20cos1|2sin0|2cosφφθθiiee0=θExample: θ = 0Superposition States as Points on the Unit Sphere|0>zyx>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψ1|1|2sin0|2cos1|2sin0|2cos0ππθθφiieeπθ=Example: θ = π, φ = 0|1>Superposition States as Points on the Unit Sphere|0>zyx2/πθ=Example: θ = π/2, φ = 021|0|1|4sin0|4cos1|2sin0|2cos0>+>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψππθθφiiee21|0| >+>|1>Superposition States as Points on the Unit SphereSuperposition States as Points on the Unit Sphere|0>zyx2/πθ=Example: θ = π/2, φ = π/221|0|1|4sin0|4cos1|2sin0|2cos2>+>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψieeiiππθθπφ21|0| >+>|1>2/πφ=21|0| >+> iTo be useful for computation, you need operations that control the state of one qubit based on the state of another. Controlled NOT or CNOT:Two-qubit operation that flips the second qubit state based on the first qubit stateinput state outputstate|0,0> |0,0>|0,1> |0,1>|1,0> |1,1>|1,1> |1,0>Example, performing a CNOT operation on α|1,1> + β|0,1> + γ |1,0>yields:α|1,0> + β|0,1> + γ |1,1>if true in macroscopic objects Quantum Entanglement (Schrodinger, 1935) Multiple quantum systems can exist in entangled super-positions of states in which the state of an individual system has no well-defined physical meaning+and(dead, live)(live, dead)Superposition and entanglement are unobservable in ordinary "macroscopic" objects due to interactions with other degrees of freedom and the surrounding world (dissipation and decoherence) … how macroscopic is too macroscopic?baba>>+>>=0|1|1|0|βαψQuantum Mechanics and Quantum ComputingnA classical computer with an n-bit memory can access 2 states. For example, with n=2 bits the 22 = 4 states are 00, 01, 10 and 11. A quantum computer can access superposition statesand entangled states. With n qubits, this gives of order states.22nExample: for n=1 qubit we can have:21 0 +=xψ11=ψ0=oψ21 i 0 −=− yψ21 i 0 +=yψExample: for n= 2 qubits we can have 36 product states such as:21 0 −=−xψ⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛+=21 021 0 ixyψ1111=ψ00=ooψ121 i 01⎟⎟⎠⎞⎜⎜⎝⎛−=yψ211 001+=eψ⎟⎟⎠⎞⎜⎜⎝⎛−=−21 00xoψ011=oψ101=oψplus entangled states (can’t be written as product) such as:211 003ie+=ψ211 002−=eψ211 i 004−=eψKey Question: can a useful quantum computer be built in practice?Answer: Definitely maybe.Main Experimental Challenge: All systems experience noise and interact with other quantum systems (the outside world), and this eventually destroys the delicate quantum superposition states. This is called decoherence.Decoherence is best understood using density matrix.Here we will just try to understand how you can manipulate the quantum state of a multi-qubit system to perform operations.The extra states can be used to tackle some very difficult tasks: - use Shor's algorithm to factor large numbers quickly and break RSA encrypted messages, - simulating other quantum systems, - efficiently searching large data-bases (Grover’s Algorithm)?|0>zyx>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψ0|1|20sin0|20cos1|2sin0|2cosφφθθiiee0=θExample: θ = 0Single qubit control operations as rotations on the Bloch sphere|0>zyx>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψ1|1|2sin0|2cos1|2sin0|2cos0ππθθφiieeπθ=Example: θ = π, φ = 0|1>starting from |0>rotate about the y-axis by π….. πy-pulse….or … NOT since such a rotation would also change |1> to |0>Single qubit control operations as rotations on the Bloch sphere|0>zyx2/πθ=Example: θ = π/2, φ = 021|0|1|4sin0|4cos1|2sin0|2cos0>+>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψππθθφiiee21|0| >+>|1>starting from |0>rotate about the y-axis by π/2… π/2-pulse….or ...since two such rotations would produce NOTNOTSingle qubit control operations as rotations on the Bloch sphere|0>zyx2/πθ=Example: θ = π/2, φ = π/221|0|1|4sin0|4cos1|2sin0|2cos2>+>=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=>⎟⎠⎞⎜⎝⎛+>⎟⎠⎞⎜⎝⎛=Ψieeiiππθθπφ21|0| >+>|1>2/πφ=starting from rotate about the z-axis by π/2. This is πZ/2 or “π/2 phase gate”since it will increase phase term for any stateby π/2”21|0| >+> i()21|0| >+>Single qubit control operations as rotations on the Bloch sphere- Consider a 2-level system with energy splitting ΔE.


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