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Quantum Computing (QC) Overview Quantum computing (QC) Overview September 2018 Dr. Sunil Dixit Technical Fellow Why is QC Important? 2 Classical Realm 3 Classical Physics Assumptions • The universe is a giant machine • All nonuniform motion and action have cause – Uniform motion does not have cause (principle of inertia) • If the state of motion is known now then all past and future states are accurately predictable because the universe is predictable • Light is a wave described completely by Maxwell’s electromagnetic equations • Waves and particles are distinct • A measurement can be accurately made and errors corrected caused by the measurement tool 4 Single Slit – Classical Marbles 5 Double Slits – Classical Marbles 6 Single Slit – Classical Waves 7 Double Slit – Classical Waves 8 Quantum Realm 9 Single Slit – Quantum Electrons 10 Double Slits – Quantum Electrons 11 Double Slit – Shoot One Electron At A Time 12 Double Slits – Quantum Electrons With Observer (Measure At One Slit) 13 Computational Capacity in the Universe • Maximum possible elementary quantum logic • Provides upper bounds computational operations: capacity performed by all matter since the • With gravitational degrees of freedom Universe began taken into account • Provides lower bounds of a quantum t computer required to simulate the entire 10120 t 2 Universe required operations and bits p • If the entire Universe performs a t 1010 years is the age of the universe computation, these numbers give the numbers of operations and bits in that with t Gh/ c5 5.391 10 44 sec p computation is Planck time (the time scale at which gravitational effects are the same order as the quantum effects) • With registered quantum fields alone: t 90 3/4 10 t p *Seth Lloyd, “Computational Capacity of the Universe”, Phys. Rev. Letters, 88(23), 2002 14 https://en.wikipedia.org/wiki/Observable_universe Quantum Computing Principles 15 Quantum Principles Important For QC • Mathematics – Primarily Linear Algebra See Backup Slides – Mathematical Notation – the Dirac Notation • Superposition • Information Representation • Uncertainty Principle • Entanglement • 6 Postulates of Quantum Mechanics 16 Quantum Superposition & Uncertainty Principle h Et 2 17 Quantum Information Representation • Physical Representation (Superposition and Entanglement) – Electrons Spin Up / Spin Down Quantum Dots – Nuclear Spins Trapped Ions Optical Lattices • Nuclear Magnetic Resonance – Polarization of Light / Photons – Optical Lattices – Semiconductor Quantum DOT – Semiconductor Josephson Junctions – Ion Traps – Others • Classical Representation – BIT (0,1) • Quantum Representation – Quantum BIT (qubit) BLOCH representation of a qubit rBLOCH rˆ x, r ˆ y , r ˆ z sin cos ,sin sin ,cos 18 Quantum Entanglement Electrons 19 Quantum Entanglement Photons 20 Quantum Entanglement (continued) • Start with 2-qubits: 0| 0 1 |1and 0 | 0 1 |1 – Both are their basis states • How do we entangle them mathematically? – Take the tensor product between the states 0| 0 1 |1 0 | 0 1 |1 0 0| 00 0 1 | 01 1 0 |10 1 1 |11 • 2-qubits in arbitrary states cannot be decomposed into their separate qubit 1 state. As an example, one of the Bell state | | 00 |11 , cannot be 2 separated into its individual qubit state • Einstein called entanglement as “spooky action at a distance,” as it appeared to violate the speed limit of information transmission in theory of relativity (i.e., “c” the velocity of light) 21 Qubit & Nuclear Spin Nuclear Magnetic Resonance 22 6 Postulates of QM deferred to backup slides 23 Basic Classical & Quantum Computer Operations & Flowchart Of Quantum Control Control of quantum operations Classical Quantum Computer Processor (Master) (Slave) Results of Measurements Input x Output y n-bit n-bit string string • Classically Quantum Mechanics y measurements only Exponential Superposition reveal n-bits of May Repeat information • Probability of 100-bit “Quantification 2 Classical Manipulate Measure string y is |αy| ; new ”- n qubits in Classical Input Apply Gates state | x Output state post transformed measurement is |y via superposition May Repeat 24 Quantum computer image from: Nature 519, 66–69 (05 March 2015) doi:10.1038/nature14270 Physical Quantum Computer D-Wave IBM 1000? D-Wave Markets 1000 qubit computers for IBM 5 Qubit Microsoft $10M - $15M 25 Quantum Computing Models 26 Models of Quantum Computing Model Description QC Circuits / Gates Adiabatic QC (Vary Hamiltonians slowly from initial to final state) s = (1/T) ˆ H(1 s ) Hinitial sH final Topological QC (World lines of particles positioned in a plane with time flowing downwards) Computational power of Anyons Measurement Based QC (Cluster States, Tomography) (local measurement is the only operation needed) 27 Quantum Circuits & Gates 28 Quantum Circuits Quantum Circuits Error Corrections Gate Graphical Mathematical Form Comments 1000 CNOT gate is a generalized XOR gate: its action 0100 on a bipartite state |A,B> is |A, B A>, where CNOT 0001 is addition modulo 2 (an XOR operation) 0010 1 0 0 0 Swaps states: ,, 0 0 1 0 SWAP 0 1 0 0 0 0 0 1 1111 H-gate (square root NOT gate) is an 2 Hadamard xz idempotent operator: H = I. It transforms the 2211 computational basis into equal superpositions. XYZ,, Quantum NOT is identical to σx => leaves |0> Pauli X, Y, Z 0 1 0i 1 0 invariant and changes the sign of |1>. ,, Rotations about the X, Y, Z axis 1 0 i 0 0 1 10 Applies a phase shift to the target qubit. i T-Gate e 4 | 00 remains same i 8 4 i 0 e e 4 |11 target qubit phase shift 1 0 0 0 Measurement collapses the superposed Measurement , quantum states 0 0 0 1 Qubit Wire = single qubit n Qubits Wire with n qubits Classical bit Double wire = single bit 29 {H, T, and CNOT} are called the “Standard Set.” Others in charts below Properties of Quantum Circuits • Are not acyclic (no loops) • No FANIN. This implies that the circuit is not reversible; does not obey unitary operation • No FANOUT. Cannot copy the qubit’s state during the computational phase – No-Cloning Theorem • No copies of qubits in superposition (produces a multipartite entangled state) | ||| |0 |1 |0 |1 |0 |1; NOT ALLOWED | 0 |1 | 0 |1 | 0 |1 | 000 |111 | ; Entangled 3qubits | | 30 IBM 5-Qubit Quantum Computer Using Toffoli Gates – Freely Available Quantum Computing 5Q SQRT(Toffoli) State Timeline for IBM 17-qubit computer is unknown 3Q Toffoli State Uses QASM (IBM Q) Assembler or QISKit SDK (Python code) discussed later, for producing the QC circuit results 31 https://quantumexperience.ng.bluemix.net/qx/editor Multi Qubit Gates (continued) 12) Qubit Teleportation Circuit An arbitrary qubit is transferred from one location to another. In literature ALICE and BOB example is commonly utilized. Teleportation takes two classical bits to one quantum state. | H Alice Space Bob | AALICE | BBOB | Squiggly lines correspond to movement of qubits. Straight reconstruct | lines correspond to movement of bits Step 4 moves from the lower left hand corner from Alice to Bob in the upper right hand corner. Time Only two classical bits remain with Alice in Step 4. Step 3 measure SINGLE QUANTUM PARTICLE IS TELEPORTED Alice sends (with speed < speed of light) the two classical bits to Bob along a classical channel. Without these Bob will not know what he has received Step 2 | ,A interact Entanglement, as well, is not transported faster than the speed of light despite its undisputable magic Infinite amount of information is passed with the qubit, however once Bob measures he can only get one bit of AB, entangled information Step 1 32 Quantum Teleportation Video Quantum-Kit Simulation: https://en.wikipedia.org/wiki/Quantum_teleportation#/media/File:Quantum_Teleportation.gif 33 Quantum Computing Programming Languages 34 QC Programming Languages and QC Simulators Product Description Website QCL C like syntax and complete. The current version of http://tph.tuwien.ac.at/~oemer/qcl.html QCL is 0.6.4 (Mar 27 2014), Source Distribution: qcl- 0.6.4 (gcc 4.7 / gnu++98 compliant), Binary Distribution (64 bit): qcl-0.6.4-x86_64-linux-gnu.tgz (AMD64, Linux 3.2, glibc2.13) QASM Assembler: Maps directly to quantum circuit model https://www.media.mit.edu/quanta/qasm2circ/ instructions https://qiskit.org/documentation/quickstart.html MIT: qasm2circ; QISKit: openQASM QISKit SDK QISKit, a quantum program is an array of quantum https://qiskit.org/documentation/quickstart.html Terra–Python circuits developed by IBM. Python program code https://github.com/QISKit/qiskit-terra API-Python workflow consists of three stages: Build, Compile, https://github.com/QISKit/qiskit-api-py and Run. Q# Is a C# like quantum programming language https://www.microsoft.com/en- developed by Microsoft. It come with a quantum us/quantum/development-kit simulator in the quantum development kit. CodeProject Is a Java quantum code project https://www.codeproject.com/Articles/1130092/Java- based-Quantum-Computing-library Quantum-Kit Is a graphical quantum circuit simulator https://sites.google.com/view/quantum-kit/home Other Simulators C/C++, CaML, Ocaml, F#, GUI based, Java, https://quantiki.org/wiki/list-qc-simulators in various Javascript, Julia, Maple, Mathematica, Maxima, languages and Matlab/Octave, .NET, Online Services, Perl/PH,P, tools Python, Rust, and Scheme/Haskell/LISP/ML Other Languages See Wikipedia https://en.wikipedia.org/wiki/Quantum_programming 35 Simple QSAM Example Using QRAM Model ۄINITIALIZE R 2 Allocates register R to 2 qubits and initializes to |00 U TENSOR H H 1 1 1 1 APPLY U R 1 1 1 1 1 UHH MEASURE R RES 2 1 1 1 1 1 1 1 1 Measures the q-register R and 1 stores in bit array RES. What is 2 the probability for the ground 1 1 1 1 1 1 state (i.e., expectation value)? 1 1 1 1 0 2 1 2 11 from coefficient of | 00 : 0.25 2 1 1 1 1 0 1 24 1 1 1 1 0 2 Parallel application of H to 1 the 2-qubits puts the register R in a balanced 2 superposition of four basis 1 1 1 1 states | 00 | 01 |10 |11 36 2 2 2 2 Quantum Algorithms 37 Quantum Computing Algorithms (continued) Algorithm Description Reference Algorithms Based on QFT Shor’s; O n2 log N 3 Integer factorization (given integer N find its Peter W.
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