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**Quantum Logic and Quantum gates with Photons**

By Simeon (Shimon) Shpiz

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Outline Intro to quantum information/computation - Qubits, their representations and quantum gates. Theoretical implementation of quantum gates. Experimental results- CNOT gates.

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**Qubit Classical bits can be either one or zero.**

Quantum bits can exist in a superposition of these two basic states. where A Qubit could possibly be implemented using any two state physical quantum system. Example states: Optics: Photon Spatial or Polarization modes. Condensed Matter: Two level atoms (Ion Traps), Magnetic Spin.

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**Models for Quantum Computation**

Adiabatic QC- this architecture works (theoretically of course) by finding a complex Hamiltonian whose ground state is a solution to the problem and then evolving a simple prepared Hamiltonian to the complex one. Cluster State QC- is an architecture in which computation evolves by making a series of single qubit measurements on a highly entangled initial state called the cluster state. Quantum Circuit – this is the quantum analogy of a Turing machine. We will focus on this model. These (and other) models for QC were shown to be computationally equivalent. Some, however, are more feasible than others.

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**The Bloch Sphere A Qubit can be represented as**

We can visualize it on the Bloch Sphere:

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Single Qubit Gates In order for a qubit to maintain its quantum state, a quantum gate has to be represented by a unitary transformation, and as such must be reversible. Examples gates: NOT gate- equivalent to Pauli X: Sign shift- equivalent to Pauli Z : Hadamard: On the Bloch sphere these gates are rotations: NOT- a 180 degree rotation about the X axis. Sign Shift- a 180 degree rotation about the Z axis Hadamard- a 90 degree rotation about the Y axis, followed by a reflection through the XY plane. Todo- ADD sigma x and sigma z

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Multiple qubit gates Multiple qubit gates can be represented as 2^N x 2^N transformations Can be constructed from the C-NOT gate with single qubit gates This means that CNOT along with single qubit rotations are a universal system- we can approximate any unitary transformation with these gates. CNOT: Conditional sign flip The CNOT gate also be implemented using a Conditional sign flip gate with two Hadamard gates (and vice versa):

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Quantum circuits Acyclic- the circuit’s output can’t be fed into its input. We can only use unitary transformation so the circuits must be reversible; i.e. no FANIN/FANOUT. No cloning – not possible to make copies of a single qubit without destroying it.

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No cloning proof Assume we want to clone our state into an arbitrary state e using a Unitary cloning operator U: This should also work for another state: By looking at the inner product of these two equations we see that: Conclusion- our states are either orthogonal (classic- 0 or 1) or identical, meaning we can’t clone an arbitrary state.

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**Parallel Computation? Consider the following scheme:**

If we have a unitary operator (representing a function) and we input a superposition of states (can be created by applying Hadamard on 0), we see that all possible outcomes are encoded in the output state. This will collapse to a single output upon measurement but the information can still be used by clever quantum algorithms to achieve classically impossible results.

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**Optical Quantum Computation**

In this scheme the qubits are photons. The qubit state is encoded in either the polarization or Spatial modes. Logical 0 Logical 1 Polarization |H> |V> Spatial |10> |01> The spatial representation is called the dual rail representation where logical 0 is represented by one photon in mode 0 and none in mode 1. Logical 1 is just the opposite.

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Optical Quantum gates Phase Shifters: Delay a photon mode by passing it through a medium with a higher refraction index. Perform a rotation around the Z axis: For polarization qubits this can be accomplished by a half-wave plate with one of the optical axis aligned with the H polarization. Beam Splitters: Transmit a fraction T and reflect (1-T) of the beam. Generally can also apply a rotation. The unitary matrix for the beam splitter is: For polarization qubits, this can be done using half wave plates. PBS(Polarization BS) : Transmits one polarization mode and reflects the other: This is used to convert from spatial to polarization encoding and vice versa .

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**Outline Intro to quantum information/computation.**

Theoretical implementation of quantum gates. Experimental results- CNOT gates.

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**Theoretical implementation of quantum gates.**

“A scheme for efficient quantum computation with linear optics ” – E. Knill , R. Laflamme & G. J. Milburn. (KLM) Results: Nondeterministic quantum computation is possible using linear optics The probability of success of a circuit can be made arbitrarily close to one.

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Nonlinear Sign Flip The following is a scheme for a Nonlinear sign flip. This means that if the mode has 2 photons then we will apply a sign flip to the amplitude. We can break down the operation of this circuit into 3 matrices.

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**Nonlinear Sign Flip If we use the parameters specified by KLM we get:**

We can write the input state as: Since the BSs and PSs are linear we can apply the U to each of the vectors separately and then multiply them. Some algebra…

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Nonlinear Sign Flip Now lets see what the chance of success of this apparatus is. Remember- we post select only the states in which we measure 1 and 0 photons in outputs 2 and 3 respectively. Case 0- zero photons in input. Case 1- one photon in input Case 2- two photons in input

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**Conditional sign flip:**

If the input is |11> this means that modes one and 3 are populated by one photon each. After the first BS they emerge bunched and head for of the NS-1 apparatuses This will add a (- ) factor to their state. In any other case the NS-1 do nothing, and we get the desired behavior. Assuming the BSs are perfect the probability of success is like that of the NS-1 squared – 1/16.

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**Destructive C-NOT gate Quantum Encoder Nondestructive CNOT**

T. B. Pittman, B. C. Jacobs, and J. D. Franson– Probabilistic quantum logic operations using polarizing beam splitters Quantum parity check Destructive C-NOT gate Quantum Encoder Nondestructive CNOT

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**Quantum parity check This transformation can be viewed as: In: Out:**

Where the last term represents states where we do not detect exactly one photon. If we only count states where we measure an F photon then we can see that the input is encoded in the out photon. This succeeds with probability of ¼ . We can improve to ½ by allowing an S photon and then applying a phase shift to the output.

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Destructive CNOT To understand the destructive CNOT we see what t does for arbitrary target and a V polarized control (for an H polarized control it is vey similar) In: Out: Where the last term represents states in which we do not measure exactly one photon. If we look at the result in HV basis we can see that : If we only count states where we measure an H photon then we can see that we have flipped the target qubit. This succeeds with probability of ¼ . We can improve to ½ by allowing an V photon and then applying a 90 degree phase shift to the output, and then imparting a polarization dependant π-phase shift as in quantum parity check. I the control is H polarized then we can see that the result will be:

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Quantum Encoder A quantum encoder encodes the state of the input into the two output qubits. It is like a parity check,only this time the a photon is part of a Bell state: In the parity check we had: We can see that for this case this translates to: With the same success probability as in the parity check- ½.

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CNOT The idea is to use the encoder to encode the control into the input of the destructive CNOT and the output- thus resulting in a non destructive CNOT gate.

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**Outline Intro to quantum information/computation.**

Theoretical implementation of quantum gates. Experimental results- CNOT gates.

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Experimental controlled-NOT logic gate for single photons in the coincidence basis 2003 T. B. Pittman, M. J. Fitch, B. C Jacobs, and J. D. Franson An attempt to build an experimental CNOT gate. A different technique was used using 1 ancilla photon in lieu of two. This was done as it is hard to produce heralded entangled pairs, while parametric down conversion allows to relatively easily produce single photons. This includes an encoder but in order to work it needs to measure all 3 photons- demolition measurement. This means that this method is not scalable but still shows the functionality.

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**Experimental CNOT- the setup**

An equally mixed 0 and 1 state is the ancilla photon (A). Let the C(control) and T(target) photons be in an arbitrary (normalized) state. Under ideal conditions the initial state is transformed into: The last term is the sum of all the amplitudes that are orthogonal to the “coincidence basis”. Now we know that if we measured 0 for A then the flip worked and if we measured 1 then we know it didn’t and we can correct that. This means that this gate can be made to work with 1/8 chance of success.

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**Experimental CNOT- Results**

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**Demonstration of an all-optical quantum controlled-NOT gate J. L**

Demonstration of an all-optical quantum controlled-NOT gate J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph & D. Branning This circuit implements a CNOT gate with probability 1/9. The beams reflect the photon with a sign change if the arrive from side marked with π. The numbers are the reflection probability. This behaves like a CNOT gate iff one photon is detected at each qubit.

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**Experimental CNOT (O’brien)-analysis**

The following equations describe the creation operators (daggers omitted): Assume the input state is an arbitrary 2 qubit state: Then the output is: If we substitute the ‘out’ operators for their values above and omit any members that don’t satisfy our constraint (one photon for each qubit) we get: Which is exactly what we want!

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**Experimental CNOT (O’brien)- the experiment**

The experimental setup is shown to the right. One can see that they ingeniously use one 1/3 wave plate for the entire process. Description: C and T are prepared. The first and last HWPs perform Hadamard, which is equivalent to a balanced BS. The first PBS convert the polarization modes into spatial modes, to allow modes from C and T to interfere and the “1/3” BS. The second HWP rotates the polarization, equivalently to a “1/3” BS in the polarization basis. The second PBS reassembles the spatial modes.

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**Experimental CNOT (O’brien)- Results**

Result summary: The gate woks well for |C>=|0> inputs (94% ± 2%,95%± 2%). This is because only one classical interference is needed to work correctly. For the |C>=|1> cases, where a non classical interference is needed the gate worked less well (75%± 2%,72%± 2%).

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Bibliography M.A. Nielsen, I.L Chuang, “Quantum computation and quantum information”, Cambridge university press, Cambridge (2000) E. Knill, R. Laflamme, G.J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) T.B. Pittman, B.C. Jacobs, J.D. Franson, “Probabilistic quantum logic using polarizing beam splitters”, Phys. Rev. A 64, (2001) J.L O’Brien, G.J. Pride, A.G. White, T.C. Ralph, D.Branning, “Demonstration of an all-optical quantum controlled-NOT gate”, Nature 426, 264 (2003) T.B. Pittman, M.J. Fitch, B.C. Jacobs, J.D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis”, Phys. Rev. A 68, (2003)

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