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 4
Single Slit – Classical Marbles 5 5
Double Slits – Classical Marbles 6 6
Single Slit – Classical Waves 7 7
Double Slit – Classical Waves 8 8
Quantum Realm 9
Single Slit – Quantum Electrons 10 10
Double Slits – Quantum Electrons 11 11
Double Slit – Shoot One Electron At A Time 12 12
Double Slits – Quantum Electrons With Observer (Measure At One Slit) 13 13
Computational Capacity in the Universe Provides upper bounds computational • • Maximum possible elementary quantum logic capacity performed by all matter since the operations: Universe began • With gravitational degrees of freedom Provides lower bounds of a quantum • taken into account computer required to simulate the entire t 120 10 Universe required operations and bits 2 t If the entire Universe performs a p • 10 computation , these numbers give the 10 years is the age of the universe t numbers of operations and bits in that 5 44 with / 5.391 10 sec t Gh c computation p 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 10 3/4 t p *Seth Lloyd, “Computational Capacity of the Universe”, Phys. Rev. Letters, 88 (23), 2002 14 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 16
Quantum Superposition & Uncertainty Principle h E t 2 17 17
Quantum Information Representation • Physical Representation (Superposition and Entanglement) – Electrons Spin Up / Spin Down Quantum Dots Optical Lattices – Nuclear Spins Trapped Ions • 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 BLOCH representation of a qubit – Quantum BIT (qubit) ˆ ˆ ˆ , , sin cos ,sin sin ,cos r r r r BLOCH x y z 18
Quantum Entanglement Electrons 19 19
Quantum Entanglement Photons 20 20
Quantum Entanglement (continued) • Start with 2-qubits: | 0 |1 | 0 |1 and 0 1 0 1 – Both are their basis states • How do we entangle them mathematically? – Take the tensor product between the states | 0 |1 | 0 |1 0 1 0 1 | 00 | 01 |10 |11 0 0 0 1 1 0 1 1 • 2-qubits in arbitrary states cannot be decomposed into their separate qubit 1 state. As an example, one of the Bell state , cannot be | | 00 |11 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 21
Qubit & Nuclear Spin Nuclear Magnetic Resonance 22 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 measurements only Quantum Mechanics y reveal n-bits of Exponential Superposition information May Repeat • Probability of 100-bit string y is |α y | 2 ; new “Quantification Classical Measure Manipulate ” - n qubits in Classical | y | x state post Input Apply Gates state Output measurement is | y transformed via superposition May Repeat Quantum computer image from: Nature 519, 66 – 69 (05 March 2015) 24 doi:10.1038/nature14270
Physical Quantum Computer D-Wave IBM 1000? D-Wave Markets 1000 IBM 5 Qubit Microsoft qubit computers for $10M - $15M 25 25
Quantum Computing Models 26
Models of Quantum Computing Model Description QC Circuits / Gates Adiabatic QC (Vary Hamiltonians slowly from initial to final state) ˆ s = (1/T) (1 ) H s H sH initial 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 H-gate (square root NOT gate) is an 1 1 1 1 idempotent operator: H 2 = I. It transforms the Hadamard x z 1 1 2 2 computational basis into equal superpositions. Quantum NOT is identical to σ x => leaves |0> X Y Z , , invariant and changes the sign of |1>. Pauli X, Y, Z 0 1 0 1 0 i Rotations about the X, Y, Z axis , , 1 0 0 0 1 i Applies a phase shift to the target qubit. 1 0 T-Gate i 4 | 00 remains same e 8 i 0 4 e i 4 |11 target qubit phase shift e Measurement collapses the superposed 1 0 0 0 quantum states Measurement , 0 0 0 1 Wire = single qubit Qubit Wire with n qubits n Qubits Double wire = single bit Classical bit {H, T, and CNOT} are called the “Standard Set.” Others in charts below 29 29
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 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 https://quantumexperience.ng.bluemix.net/qx/editor 31 31
Multi Qubit Gates (continued) An arbitrary qubit is transferred from one location to another. In 12) Qubit Teleportation Circuit literature ALICE and BOB example is commonly utilized. | Teleportation takes two classical bits to one quantum state. H Space Bob Alice | A ALICE | | | B BOB reconstruct | Squiggly lines correspond to movement of qubits. Straight 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. measure Step 3 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 | , interact A Step 2 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 | , entangled A B Step 1 information 32
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