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Quantum Computing UfUf H H H Nick Bonesteel Discovering Physics, Nov. 16, 2012

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What is a quantum computer, and what can we do with one?

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A Classical Bit: Two Possible States 0

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1

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0 x y NOT x y Single Bit Operation: NOT

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1 x y NOT x y Single Bit Operation: NOT

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A Quantum Bit or “Qubit” 0

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1

9 0

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01

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01 Quantum superposition of 0 and 1 A Quantum Bit or “Qubit”

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1

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01

14 0

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A Quantum Bit: A Continuum of States sin cos 0 1

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A Quantum Bit: A Continuum of States 1 2 sin0 2 cos i e Actually, qubit states live on the surface of a sphere.

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A Quantum Bit: A Continuum of States sin cos 0 1 But the circle is enough for us today.

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A Quantum NOT Gate X

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XX

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X

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XX

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Hadamard Gate HH

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HH

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HH

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HH

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HH H is its own inverse

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Hadamard Gate H is its own inverse HH

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Hadamard Gate HH H is its own inverse

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Hadamard Gate H is its own inverse HH

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Fair Coin Trick Coin

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Balanced Function Unbalanced Function or

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UfUf A Two Qubit Subroutine to Evaluate f(x)

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UfUf Input x can be either 0 or 1 Output is f(x) Initialize to state “0”

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A Two Qubit Subroutine to Evaluate f(x) UfUf Input x can be either 0 or 1 This qubit can also be in state “1”

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UfUf Bar stands for “NOT” 0 = 1, 1 = 0 A Two Qubit Subroutine to Evaluate f(x) Input x can be either 0 or 1 This qubit can also be in state “1”

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UfUf UfUf Balanced Unbalanced or A Two Qubit Subroutine to Evaluate f(x)

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A Quantum Algorithm (Deutsch-Jozsa ‘92) UfUf H H H

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UfUf H H H

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UfUf H H H

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UfUf H H H

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UfUf H H H

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UfUf H H H Only ran U f subroutine once, but f(0) and f(1) both appear in the state of the computer! A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is balanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is balanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is balanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is unbalanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is unbalanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H If f is unbalanced: f(0) = f(1) and f(0) = f(1) A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H Unbalanced: Balanced: A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H Unbalanced: Balanced: A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H Unbalanced: Balanced: Measure top qubit A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H Unbalanced: Balanced: Measure top qubit A Quantum Algorithm (Deutsch-Jozsa ‘92)

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UfUf H H H Unbalanced: Balanced: Measure top qubit A Quantum Algorithm (Deutsch-Jozsa ‘92)

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One qubit H

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HH Two qubits

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HH Counting in binary Two qubits

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HHH Three qubits

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HHH

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HHH H N N -1 … N qubits

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HHH H Quantum superposition of all possible input states! N qubits

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HHH H Quantum superposition of all possible input states! For N=250 the number of states is roughly the number of atoms in the universe! N qubits

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HHH H Quantum superposition of all possible input states! For N=250 the number of states is roughly the number of atoms in the universe! UfUf One function call N qubits

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HHH H Quantum superposition of all possible input states! For N=250 the number of states is roughly the number of atoms in the universe! UfUf One function call x can be any integer From 0 to 2 N -1 N qubits

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HHH H Quantum superposition of all possible input states! UfUf One function call Evaluate f(x) for all possible inputs! x can be any integer From 0 to 2 N -1 N qubits

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Massive Quantum Parallelism H H H H UfUf Only one problem: When I measure this state I only learn the value of f(x) for one input x. (No free lunch!) However, people have shown that a quantum computer can use quantum parallelism to do things no classical computer can do. Run program U f once, get result for all possible inputs!

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Given two prime numbers p and q, p x q = C Easy C p, q Hard Best known classical factoring algorithm scales as time = exp(Number of Digits) Mathematical Basis for Public Key Cryptography. Prime Factorization

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In 1994 Peter Shor showed that a Quantum Computer could factor an integer exponentially faster than a classical computer! time = (Number of Digits) Shor’s algorithm exploits Massive Quantum Parallelism. 3 Quantum Factorization

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OK, so how do we make a quantum computer?

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Boolean Logic Gates x y zx y NotNOR Any classical computation can be carried out using these two gates x y x y z

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Transistor Logic A A A B A B

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The Integrated Circuit Core i7: 731,000,000 transistors

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Single Qubit Gates U

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X X Controlled-NOT Gate X X

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X X X X U U Any quantum computation can be carried out using these two gates Universal Set of Gates U X Quantum Circuit

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2012 Nobel Prize in Physics Dave Wineland Serge Haroche

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State of the Art: Superconducting Qubits From : “Quantum Computers,” T. D. Ladd et al., Nature 464, (2010)

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High fidelity (~95%) 2-qubit gates on a time scale of 30 ns. Nature 460, (2009) 2 superconducting qubits coupled by a microwave resonator

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Nature 460, (2009) 2 superconducting qubits coupled by a microwave resonator High fidelity (~95%) 2-qubit gates on a time scale of 30 ns.

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8 mm First steps toward a scalable quantum computer From:

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The Real Problem: Decoherence!

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Qubit

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The Real Problem: Decoherence! Qubit Everything else

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The Real Problem: Decoherence! Qubit Everything else Over time….

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The Real Problem: Decoherence! Qubit Everything else Over time…. Quantum coherence of qubit is inevitably lost!

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The Real Problem: Decoherence! Qubit Everything else Over time…. Quantum coherence of qubit is inevitably lost! Amazingly enough, quantum computing is still possible using what is known as “fault-tolerant quantum computation.”

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Coherence Times for Superconducting Qubits From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011) Threshold for fault- tolerant quantum computation.

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From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011) Threshold for fault- tolerant quantum computation. This is what most of my own research on quantum computing is focused on. Coherence Times for Superconducting Qubits

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A deep question: Do the laws of nature allow us to manipulate quantum systems with enough accuracy to build “quantum machines” ? Conclusions

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A deep question: Do the laws of nature allow us to manipulate quantum systems with enough accuracy to build “quantum machines” ? If the answer is “yes” (as it seems to be), then quantum computers are coming! Conclusions

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