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Introduction to Quantum Computing Lecture 1 of 2
CS 497 Frontiers of Computer Science Introduction to Quantum Computing Lecture 1 of 2 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo January 16 & 18, 2007
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Moore’s Law Following trend … atomic scale in 15-20 years
1975 1980 1985 1990 1995 2000 2005 104 105 106 107 108 109 number of transistors year Following trend … atomic scale in years Quantum mechanical effects occur at this scale: Measuring a state (e.g. position) disturbs it Quantum systems sometimes seem to behave as if they are in several states at once Different evolutions can interfere with each other
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Quantum mechanical effects
Additional nuisances to overcome? or New types of behavior to make use of? [Shor ’94]: polynomial-time algorithm for factoring integers on a quantum computer This could be used to break most of the existing public-key cryptosystems, including RSA, and elliptic curve crypto [Bennett, Brassard ’84]: provably secure codes with short keys
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Also with quantum information:
Faster algorithms for combinatorial search problems Fast algorithms for simulating quantum mechanics Communication savings in distributed systems More efficient notions of “proof systems” classical information theory quantum information Quantum information theory is a generalization of the classical information theory that we all know—which is based on probability theory
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Classical and quantum systems
Probabilistic states: Quantum states: Dirac notation: |000, |001, |010, …, |111 are basis vectors, so
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Dirac bra/ket notation
Ket: ψ always denotes a column vector, e.g. Convention: Bra: ψ always denotes a row vector that is the conjugate transpose of ψ, e.g. [ *1 *2 *d ] Bracket: φψ denotes φψ, the inner product of φ and ψ
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Basic operations on qubits (I)
Recall (0) Initialize qubit to |0 or to |1 (1) Apply a unitary operation U (formally U†U = I ) conjugate transpose Examples: Rotation by : NOT (bit flip): Maps |0 |1 |1 |0 Phase flip: Maps |0 |0 |1 |1
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Basic operations on qubits (II)
1 More examples of unitary operations: (unitary rotation) H0 H1 Reflection about this line Hadamard: 0
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Basic operations on qubits (III)
1 ψ1 (3) Apply a “standard” measurement: 0 + 1 ψ0 ||2 0 ||2 … and the quantum state collapses to 0 or 1 () There exist other quantum operations, but they can all be “simulated” by the aforementioned types Example: measurement with respect to a different orthonormal basis {ψ0, ψ1}
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Distinguishing between two states
Let be in state or Question 1: can we distinguish between the two cases? Distinguishing procedure: apply H measure This works because H + = 0 and H − = 1 Question 2: can we distinguish between 0 and +? Since they’re not orthogonal, they cannot be perfectly distinguished … but statistical difference is detectable
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Operations on n-qubit states
Unitary operations: (U†U = I ) Measurements: … and the quantum state collapses
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Entanglement 2. ? ?? Suppose that two qubits are in states:
The state of the combined system their tensor product: Question: what are the states of the individual qubits for 1. ? ? Answers: 2. ?? ... this is an entangled state
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Structure among subsystems
qubits: #2 #1 #4 #3 time V U W unitary operations measurements
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Quantum circuits 0 1 1 Computation is “feasible” if circuit-size scales polynomially
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Example of a one-qubit gate applied to a two-qubit system
(do nothing) The resulting 4x4 matrix is 00 0U0 01 0U1 10 1U0 11 1U1 Maps basis states as: Question: what happens if U is applied to the first qubit?
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Controlled-U gates U Resulting 4x4 matrix is controlled-U =
Maps basis states as: 00 00 01 01 10 1U0 11 1U1
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Controlled-NOT (CNOT)
X a b ab ≡ Note: “control” qubit may change on some input states! 0 + 1 0 − 1 H
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Multiplication problem
Input: two n-bit numbers (e.g. 101 and 111) Output: their product (e.g ) “Grade school” algorithm takes O(n2) steps Best currently-known classical algorithm costs O(n log n loglog n) Best currently-known quantum method: same
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Factoring problem Input: an n-bit number (e.g. 100011)
Output: their product (e.g. 101, 111) Trial division costs 2n/2 Best currently-known classical algorithm costs O(2n⅓ log⅔ n ) Hardness of factoring is the basis of the security of many cryptosystems (e.g. RSA) Shor’s quantum algorithm costs n [ O(n2 log n loglog n) ] Implementation would break RSA and other cryptosystems
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How do quantum algorithms work?
Given a polynomial-time classical algorithm for f :{0,1}n → T, it is straightforward to construct a quantum algorithm that creates the state: This is not performing “exponentially many computations at polynomial cost” The most straightforward way of extracting information from the state yields just (x, f (x)) for a random x{0,1}n But we can make some interesting tradeoffs: instead of learning about any (x, f (x)) point, one can learn something about a global property of f
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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Deutsch’s problem f f1(x) f2(x) f3(x) f4(x) Let f : {0,1} → {0,1}
There are four possibilities: x f1(x) 1 x f2(x) 1 x f3(x) 1 x f4(x) 1 Goal: determine f(0) f(1) Any classical method requires two queries What about a quantum method?
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Reversible black box for f
Uf f alternate notation: b b f(a) A classical algorithm: (still requires 2 queries) f 1 f(0) f(1) 2 queries + 1 auxiliary operation
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Quantum algorithm for Deutsch
1 2 3 0 H f H f(0) f(1) 1 H 1 query + 4 auxiliary operations How does this algorithm work? Each of the three H operations can be seen as playing a different role ...
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Quantum algorithm (1) H f f 1 0
2 3 H f 1 0 1 1. Creates the state 0 – 1, which is an eigenvector of NOT with eigenvalue –1 I with eigenvalue +1 This causes f to induce a phase shift of (–1) f(x) to x f 0 – 1 x (–1) f(x)x
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Quantum algorithm (2) H f f1(x) f2(x) f3(x) f4(x)
2. Causes f to be queried in superposition (at 0 + 1) f 0 – 1 0 (–1) f(0)0 + (–1) f(1)1 H x f1(x) 1 x f2(x) 1 x f3(x) 1 x f4(x) 1 (0 + 1) (0 – 1)
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Quantum algorithm (3) 3. Distinguishes between (0 + 1) and (0 – 1) H (0 + 1) 0 (0 – 1) 1
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Summary of Deutsch’s algorithm
Makes only one query, whereas two are needed classically produces superpositions of inputs to f : 0 + 1 extracts phase differences from (–1) f(0)0 + (–1) f(1)1 H f 1 0 f(0) f(1) 1 2 3 constructs eigenvector so f-queries induce phases: x (–1) f(x)x
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Contents of lecture 1 Preliminary remarks Quantum states
Unitary operations & measurements Subsystem structure & quantum circuit diagrams Introductory remarks about quantum algorithms Deutsch’s parity algorithm One-out-of-four search algorithm
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One-out-of-four search
Let f : {0,1}2 → {0,1} have the property that there is exactly one x {0,1}2 for which f (x) = 1 Four possibilities: x f00(x) 00 01 10 11 1 x f01(x) 00 01 10 11 1 x f10(x) 00 01 10 11 1 x f11(x) 00 01 10 11 1 Goal: find x {0,1}2 for which f (x) = 1 What is the minimum number of queries classically? ____ Quantumly? ____
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Quantum algorithm (I) f f Black box for 1-4 search: x1 x2 y
y f(x1,x2) Start by creating phases in superposition of all inputs to f: f H 1 0 Input state to query? (00 + 01 + 10 + 11)(0 – 1) Output state of query? ((–1) f(00)00 + (–1) f(01)01 + (–1) f(10)10 + (–1) f(11)11)(0 – 1)
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Quantum algorithm (II)
Apply the U that maps ψ00, ψ01, ψ10, ψ11 to 00, 01, 10, 11 (resp.) f H 1 0 U Output state of the first two qubits in the four cases: Case of f00? ψ00 = – 00 + 01 + 10 + 11 Case of f01? ψ01 = + 00 – 01 + 10 + 11 Case of f10? ψ10 = + 00 + 01 – 10 + 11 Case of f11? ψ11 = + 00 + 01 + 10 – 11 What noteworthy property do these states have? Orthogonal!
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THE END
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