Day 3 Selected Solutions
|
|
- Charla Anthony
- 5 years ago
- Views:
Transcription
1 Solutions for Exercises 1 and 2
2 Exercise 1 Start with something simple: What is the stabilizer S such that is stabilized by S? G 1 = linspan ( 00, 11 )
3 Exercise 1 Start with something simple: What is the stabilizer S such that is stabilized by S? S = Z Z. G 1 = linspan ( 00, 11 )
4 Exercise 1 Start with something simple: What is the stabilizer S such that G 1 = linspan ( 00, 11 ) is stabilized by S? S = Z Z. Now what are the stabilizers S 1, S 2 such that G 2 = linspan ( 000, 111 ) is stabilized by S 1 and S 2?
5 Exercise 1 Start with something simple: What is the stabilizer S such that G 1 = linspan ( 00, 11 ) is stabilized by S? S = Z Z. Now what are the stabilizers S 1, S 2 such that G 2 = linspan ( 000, 111 ) is stabilized by S 1 and S 2? Ensure that qubits 1 and 2 are the same and also qubits 2 and 3 are the same. S 1 = Z Z I, S 2 = I Z Z, X = X X X
6 Exercise 1 Start with something simple: What is the stabilizer S such that G 1 = linspan ( 00, 11 ) is stabilized by S? S = Z Z. Now what are the stabilizers S 1, S 2 such that G 2 = linspan ( 000, 111 ) is stabilized by S 1 and S 2? Ensure that qubits 1 and 2 are the same and also qubits 2 and 3 are the same. S 1 = Z Z I, S 2 = I Z Z, X = X X X This is the 3 qubit code resistent to X errors.
7 Exercise 1 Similarly, the 3 qubit code resistent to Z errors will have stabilizers S 3, S 4 that stabilize the space Analagously, G 3 = linspan ( + + +, ). S 3 = X X I, S 4 = I X X, Z = Z Z Z We can see this by noticing G 3 = (H H H)G 2. So the stabilizer S 3 = (HHH)(ZZI )(HHH) = XXI
8 Exercise 1 This would also work if we worked with logical qubits instead of qubits. So, when we apply the 3 qubit Z-resistent code to the 3 qubit X -resistent code, S 3 =X X I = XXXXXXIII S 4 =I X X = IIIXXXXXX Z =ZZZ = ZZZZZZZZZ
9 Exercise 1 So now we need to add the stabilizers for each of the X -resistent codes to get the entire set of stabilizers. S 1 = ZZI III III S 2 = IZZ III III S 3 = III ZZI III S 4 = III IZZ III S 5 = III III ZZI S 6 = III III IZZ S 7 = XXX XXX III S 8 = III XXX XXX X = XXX XXX XXX Z = ZZZ ZZZ ZZZ
10 Exercise 2 Simulating one qubit measurements with constant size quantum circuit We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone.
11 Exercise 2 Simulating one qubit measurements with constant size quantum circuit We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone. x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
12 Exercise 2 Simulating one qubit measurements with constant size quantum circuit We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone. x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
13 Exercise 2 Simulating one qubit measurements with constant size quantum circuit We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone. x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
14 Exercise 2 Simulating one qubit measurements with constant size quantum circuit x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone. Answer: It depends on at most D = 2 d qubits.
15 Exercise 2 Simulating one qubit measurements with constant size quantum circuit x 0 x 1 x 2 x 3 x 4 x 5 We have a depth-d quantum circuit, and we re interested in a measurement just on one qubit of the final state. How much of the circuit does this depend on? Idea: backward lightcone. Answer: It depends on at most D = 2 d qubits. This circuit has depth d and acts on D qubits. It has size at most Dd. The measurement outcome has the same distribution as the same measurement in the original circuit.
16 Exercise 2 Characterizing a constant size quantum circuit with a constant depth classical circuit Our circuit of size Dd is fully described by a sequence of matrix multiplications, where each matrix has size 2 D 2 D. Notice that the depth of the new circuit 2 D = 2 2d is doubly exponential in the depth of the original circuit!
17 Exercise 2 Characterizing a constant size quantum circuit with a constant depth classical circuit Our circuit of size Dd is fully described by a sequence of matrix multiplications, where each matrix has size 2 D 2 D. Notice that the depth of the new circuit 2 D = 2 2d is doubly exponential in the depth of the original circuit! Let be the unitary describing the circuit. Then we can compute matrix entries of in constant time using the above.
18 Exercise 2 Characterizing a constant size quantum circuit with a constant depth classical circuit Our circuit of size Dd is fully described by a sequence of matrix multiplications, where each matrix has size 2 D 2 D. Notice that the depth of the new circuit 2 D = 2 2d is doubly exponential in the depth of the original circuit! Let be the unitary describing the circuit. Then we can compute matrix entries of in constant time using the above. We can compute the probabilities of the measurement outcome like so: ( 0 I )C x D 2 = 0z C x D 2. (1) z {0,1} D 1 Each complex number 0z C x D can be computed in constant time, and we need to compute only a constant number of them. So we can compute the probability of measuring 0.
19 Exercise 2 Classical constant depth circuits compute the same functions as quantum ones Suppose that computes f most of the time, in the sense that f (x) x
20 Exercise 2 Classical constant depth circuits compute the same functions as quantum ones Suppose that computes f most of the time, in the sense that f (x) x For the i th bit, the probability that measuring the i th bit of x gives the i th bit of f (x) is at least 1 2.
21 Exercise 2 Classical constant depth circuits compute the same functions as quantum ones Suppose that computes f most of the time, in the sense that f (x) x For the i th bit, the probability that measuring the i th bit of x gives the i th bit of f (x) is at least 1 2. Therefore, we can extract each bit of the output of f.
22 Exercise 2 Classical constant depth circuits compute the same functions as quantum ones Suppose that computes f most of the time, in the sense that f (x) x For the i th bit, the probability that measuring the i th bit of x gives the i th bit of f (x) is at least 1 2. Therefore, we can extract each bit of the output of f. Theorem If quantum circuit of constant depth computes f with probability 1 2, then there is a constant-depth classical circuit computing f.
23 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j )
24 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j ) = D 1 ij i i A B j j
25 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j ) = D 1 ij = D 1 ij i i A B j j i A j i B j
26 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j ) = D 1 ij = D 1 ij = D 1 ij i i A B j j i A j i B j i A j j B T i
27 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j ) = D 1 ij = D 1 ij = D 1 ij = D 1 i i i A B j j i A j i B j i A j j B T i i A j j B T i j
28 Exercise 7 Computational lemmas Let φ D = D 1/2 i i i. φ D A B φ D = (D 1/2 i i i )A B(D 1/2 j j j ) = D 1 ij = D 1 ij = D 1 ij = D 1 i i i A B j j i A j i B j i A j j B T i i A j j B T i j = D 1 i i AB T i = D 1 Tr AB T.
29 Exercise 7 Computational lemmas Suppose AB = BA. Then also AB T = B T A.
30 Exercise 7 Computational lemmas Suppose AB = BA. Then also AB T = B T A. By cyclicity, Tr AB = Tr BA. But by linearity, Tr AB = Tr BA. Therefore, Tr AB = 0.
31 Exercise 7 Computational lemmas Suppose AB = BA. Then also AB T = B T A. By cyclicity, Tr AB = Tr BA. But by linearity, Tr AB = Tr BA. Therefore, Tr AB = 0. We also note that AB = BA iff ABA 1 B 1 = I. The quantity ABA 1 B 1 is known as the group commutator of A and B.
32 Exercise 7 Computational lemmas Suppose AB = BA. Then also AB T = B T A. By cyclicity, Tr AB = Tr BA. But by linearity, Tr AB = Tr BA. Therefore, Tr AB = 0. We also note that AB = BA iff ABA 1 B 1 = I. The quantity ABA 1 B 1 is known as the group commutator of A and B. Notice that the group commutator factorizes across the tensor product. More precisely, suppose A = A 0 A 1, B = B 0 B 1. Then ABA 1 B 1 = A 0 B 0 A 1 0 B 1 0 A 1 B 1 A 1 1 B 1 1. (2)
33 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z
34 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z We claim that C i C j = C j C i when i j.
35 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z We claim that C i C j = C j C i when i j. To see this, consider the group commutator term-by-term in the tensor product.
36 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z We claim that C i C j = C j C i when i j. To see this, consider the group commutator term-by-term in the tensor product. Exactly one group commutator is between an X and Z. The rest are between X and X or between I and something.
37 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z We claim that C i C j = C j C i when i j. To see this, consider the group commutator term-by-term in the tensor product. Exactly one group commutator is between an X and Z. The rest are between X and X or between I and something. The commutators of the latter type are all equal to I. The X, Z commutator is equal to I.
38 Exercise 7 Constructing the operators Consider the matrices C 1 = Z I I I C 2 = X Z I I C 3 = X X Z I. C d = X X X Z We claim that C i C j = C j C i when i j. To see this, consider the group commutator term-by-term in the tensor product. Exactly one group commutator is between an X and Z. The rest are between X and X or between I and something. The commutators of the latter type are all equal to I. The X, Z commutator is equal to I. Therefore, the whole commutator is I.
39 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d.
40 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d. By our first computational lemma, this is equal to 1 D Tr C(u)C(v)T.
41 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d. By our first computational lemma, this is equal to 1 D Tr C(u)C(v)T. By linearity, this is equal to 1 D ij Tr u iv j C i Cj T.
42 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d. By our first computational lemma, this is equal to 1 D Tr C(u)C(v)T. By linearity, this is equal to 1 D ij Tr u iv j C i C T By our second computational lemma, Tr C i C T j also 1 D Tr C ic T i = 1. j. = 0 for i j. But
43 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d. By our first computational lemma, this is equal to 1 D Tr C(u)C(v)T. By linearity, this is equal to 1 D ij Tr u iv j C i C T By our second computational lemma, Tr C i C T j also 1 D Tr C ic T i = 1. Therefore, φ D C(u) C(v) φ D = i u iv i = u v. j. = 0 for i j. But
44 Exercise 7 Analyzing the operators Given vector u R d, let C(u) = i u ic i. Let s compute φ d C(u) C(v) φ d. By our first computational lemma, this is equal to 1 D Tr C(u)C(v)T. By linearity, this is equal to 1 D ij Tr u iv j C i C T By our second computational lemma, Tr C i Cj T = 0 for i j. But also 1 D Tr C ici T = 1. Therefore, φ D C(u) C(v) φ D = i u iv i = u v. To conclude: any correlation achieved by inner products of vectors is also achieved by making measurements on a state. j.
SUPERDENSE CODING AND QUANTUM TELEPORTATION
SUPERDENSE CODING AND QUANTUM TELEPORTATION YAQIAO LI This note tries to rephrase mathematically superdense coding and quantum teleportation explained in [] Section.3 and.3.7, respectively (as if I understood
More informationRelations for 2-qubit Clifford+T operator group
Relations for -qubit Clifford+T operator group Peter Selinger and iaoning Bian Dalhousie University Contents Some background The main theorem Proof of the main theorem Greylyn s theorem Presentation of
More informationShort Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
More informationX row 1 X row 2, X row 2 X row 3, Z col 1 Z col 2, Z col 2 Z col 3,
1 Ph 219c/CS 219c Exercises Due: Thursday 9 March 2017.1 A cleaning lemma for CSS codes In class we proved the cleaning lemma for stabilizer codes, which says the following: For an [[n, k]] stabilizer
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationCSE 599d - Quantum Computing Stabilizer Quantum Error Correcting Codes
CSE 599d - Quantum Computing Stabilizer Quantum Error Correcting Codes Dave Bacon Department of Computer Science & Engineering, University of Washington In the last lecture we learned of the quantum error
More informationVector Space Examples Math 203 Spring 2014 Myers. Example: S = set of all doubly infinite sequences of real numbers = {{y k } : k Z, y k R}
Vector Space Examples Math 203 Spring 2014 Myers Example: S = set of all doubly infinite sequences of real numbers = {{y k } : k Z, y k R} Define {y k } + {z k } = {y k + z k } and c{y k } = {cy k }. Example:
More informationC/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9
C/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9 1 Readings Benenti, Casati, and Strini: Quantum Gates Ch. 3.2-3.4 Universality Ch. 3.5-3.6 2 Quantum Gates Continuing
More informationPrinciples of Quantum Mechanics Pt. 2
Principles of Quantum Mechanics Pt. 2 PHYS 500 - Southern Illinois University February 9, 2017 PHYS 500 - Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, 2017 1 / 13 The
More informationLOWELL. MICHIGAN. WEDNESDAY, FEBRUARY NUMllEE 33, Chicago. >::»«ad 0:30am, " 16.n«l w 00 ptn Jaekten,.'''4snd4:4(>a tii, ijilwopa
4/X6 X 896 & # 98 # 4 $2 q $ 8 8 $ 8 6 8 2 8 8 2 2 4 2 4 X q q!< Q 48 8 8 X 4 # 8 & q 4 ) / X & & & Q!! & & )! 2 ) & / / ;) Q & & 8 )
More informationQUANTUM COMPUTATION. Exercise sheet 1. Ashley Montanaro, University of Bristol H Z U = 1 2
School of Mathematics Spring 017 QUANTUM COMPUTATION Exercise sheet 1 Ashley Montanaro, University of Bristol ashley.montanaro@bristol.ac.uk 1. The quantum circuit model. (a) Consider the following quantum
More informationQuantum Computing Lecture 3. Principles of Quantum Mechanics. Anuj Dawar
Quantum Computing Lecture 3 Principles of Quantum Mechanics Anuj Dawar What is Quantum Mechanics? Quantum Mechanics is a framework for the development of physical theories. It is not itself a physical
More informationQuantum Phase Estimation using Multivalued Logic
Quantum Phase Estimation using Multivalued Logic Agenda Importance of Quantum Phase Estimation (QPE) QPE using binary logic QPE using MVL Performance Requirements Salient features Conclusion Introduction
More informationKevin James. MTHSC 412 Section 3.1 Definition and Examples of Rings
MTHSC 412 Section 3.1 Definition and Examples of Rings A ring R is a nonempty set R together with two binary operations (usually written as addition and multiplication) that satisfy the following axioms.
More informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More informationQuantum Computing. 6. Quantum Computer Architecture 7. Quantum Computers and Complexity
Quantum Computing 1. Quantum States and Quantum Gates 2. Multiple Qubits and Entangled States 3. Quantum Gate Arrays 4. Quantum Parallelism 5. Examples of Quantum Algorithms 1. Grover s Unstructured Search
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationQuantum Information & Quantum Computing
Math 478, Phys 478, CS4803, February 9, 006 1 Georgia Tech Math, Physics & Computing Math 478, Phys 478, CS4803 Quantum Information & Quantum Computing Problems Set 1 Due February 9, 006 Part I : 1. Read
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More informationC/CS/Phys 191 Quantum Gates and Universality 9/22/05 Fall 2005 Lecture 8. a b b d. w. Therefore, U preserves norms and angles (up to sign).
C/CS/Phys 191 Quantum Gates and Universality 9//05 Fall 005 Lecture 8 1 Readings Benenti, Casati, and Strini: Classical circuits and computation Ch.1.,.6 Quantum Gates Ch. 3.-3.4 Universality Ch. 3.5-3.6
More informationLecture 2: Linear operators
Lecture 2: Linear operators Rajat Mittal IIT Kanpur The mathematical formulation of Quantum computing requires vector spaces and linear operators So, we need to be comfortable with linear algebra to study
More informationSingle qubit + CNOT gates
Lecture 6 Universal quantum gates Single qubit + CNOT gates Single qubit and CNOT gates together can be used to implement an arbitrary twolevel unitary operation on the state space of n qubits. Suppose
More informationDecay of the Singlet Conversion Probability in One Dimensional Quantum Networks
Decay of the Singlet Conversion Probability in One Dimensional Quantum Networks Scott Hottovy shottovy@math.arizona.edu Advised by: Dr. Janek Wehr University of Arizona Applied Mathematics December 18,
More information1 Readings. 2 Unitary Operators. C/CS/Phys C191 Unitaries and Quantum Gates 9/22/09 Fall 2009 Lecture 8
C/CS/Phys C191 Unitaries and Quantum Gates 9//09 Fall 009 Lecture 8 1 Readings Benenti, Casati, and Strini: Classical circuits and computation Ch.1.,.6 Quantum Gates Ch. 3.-3.4 Kaye et al: Ch. 1.1-1.5,
More informationIntroduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871
Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871 Lecture 9 (2017) Jon Yard QNC 3126 jyard@uwaterloo.ca http://math.uwaterloo.ca/~jyard/qic710 1 More state distinguishing
More informationQuantum expanders from any classical Cayley graph expander
Quantum expanders from any classical Cayley graph expander arxiv:0709.1142 Aram Harrow (Bristol) QIP 08 19 Dec 2007 outline Main result. Definitions. Proof of main result. Applying the recipe: examples
More informationDeutsch s Algorithm. Dimitrios I. Myrisiotis. MPLA CoReLab. April 26, / 70
Deutsch s Algorithm Dimitrios I. Myrisiotis MPLA CoReLab April 6, 017 1 / 70 Contents Introduction Preliminaries The Algorithm Misc. / 70 Introduction 3 / 70 Introduction Figure : David Deutsch (1953 ).
More informationQuantum Algorithms. Andreas Klappenecker Texas A&M University. Lecture notes of a course given in Spring Preliminary draft.
Quantum Algorithms Andreas Klappenecker Texas A&M University Lecture notes of a course given in Spring 003. Preliminary draft. c 003 by Andreas Klappenecker. All rights reserved. Preface Quantum computing
More informationProblem Set # 6 Solutions
Id: hw.tex,v 1.4 009/0/09 04:31:40 ike Exp 1 MIT.111/8.411/6.898/18.435 Quantum Information Science I Fall, 010 Sam Ocko October 6, 010 1. Building controlled-u from U (a) Problem Set # 6 Solutions FIG.
More informationQuantum Computing Lecture Notes, Extra Chapter. Hidden Subgroup Problem
Quantum Computing Lecture Notes, Extra Chapter Hidden Subgroup Problem Ronald de Wolf 1 Hidden Subgroup Problem 1.1 Group theory reminder A group G consists of a set of elements (which is usually denoted
More informationIntroduction to Quantum Error Correction
Introduction to Quantum Error Correction Nielsen & Chuang Quantum Information and Quantum Computation, CUP 2000, Ch. 10 Gottesman quant-ph/0004072 Steane quant-ph/0304016 Gottesman quant-ph/9903099 Errors
More information)j > Riley Tipton Perry University of New South Wales, Australia. World Scientific CHENNAI
Riley Tipton Perry University of New South Wales, Australia )j > World Scientific NEW JERSEY LONDON. SINGAPORE BEIJING SHANSHAI HONG K0N6 TAIPEI» CHENNAI Contents Acknowledgments xi 1. Introduction 1 1.1
More informationThe Solovay-Kitaev theorem
The Solovay-Kitaev theorem Maris Ozols December 10, 009 1 Introduction There are several accounts of the Solovay-Kitaev theorem available [K97, NC00, KSV0, DN05]. I chose to base my report on [NC00], since
More informationSemiconductors: Applications in spintronics and quantum computation. Tatiana G. Rappoport Advanced Summer School Cinvestav 2005
Semiconductors: Applications in spintronics and quantum computation Advanced Summer School 1 I. Background II. Spintronics Spin generation (magnetic semiconductors) Spin detection III. Spintronics - electron
More informationLecture 2: Introduction to Quantum Mechanics
CMSC 49: Introduction to Quantum Computation Fall 5, Virginia Commonwealth University Sevag Gharibian Lecture : Introduction to Quantum Mechanics...the paradox is only a conflict between reality and your
More informationOptimal Realizations of Controlled Unitary Gates
Optimal Realizations of Controlled nitary Gates Guang Song and Andreas Klappenecker Department of Computer Science Texas A&M niversity College Station, TX 77843-3112 {gsong,klappi}@cs.tamu.edu Abstract
More informationLogic gates. Quantum logic gates. α β 0 1 X = 1 0. Quantum NOT gate (X gate) Classical NOT gate NOT A. Matrix form representation
Quantum logic gates Logic gates Classical NOT gate Quantum NOT gate (X gate) A NOT A α 0 + β 1 X α 1 + β 0 A N O T A 0 1 1 0 Matrix form representation 0 1 X = 1 0 The only non-trivial single bit gate
More informationIntroduction to Quantum Information Processing
Introduction to Quantum Information Processing Lecture 6 Richard Cleve Overview of Lecture 6 Continuation of teleportation Computation and some basic complexity classes Simple quantum algorithms in the
More informationAn Introduction to Quantum Computation and Quantum Information
An to and Graduate Group in Applied Math University of California, Davis March 13, 009 A bit of history Benioff 198 : First paper published mentioning quantum computing Feynman 198 : Use a quantum computer
More informationMinimal-memory, non-catastrophic, polynomial-depth quantum convolutional encoders
1 Minimal-memory, non-catastrophic, polynomial-depth quantum convolutional encoders Monireh Houshmand, Saied Hosseini-Khayat, and Mark M Wilde arxiv:11050649v4 [quant-ph] 12 Aug 2012 Abstract Quantum convolutional
More informationLecture: Quantum Information
Lecture: Quantum Information Transcribed by: Crystal Noel and Da An (Chi Chi) November 10, 016 1 Final Proect Information Find an issue related to class you are interested in and either: read some papers
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationChapter 2. Basic Principles of Quantum mechanics
Chapter 2. Basic Principles of Quantum mechanics In this chapter we introduce basic principles of the quantum mechanics. Quantum computers are based on the principles of the quantum mechanics. In the classical
More informationMonoids. Definition: A binary operation on a set M is a function : M M M. Examples:
Monoids Definition: A binary operation on a set M is a function : M M M. If : M M M, we say that is well defined on M or equivalently, that M is closed under the operation. Examples: Definition: A monoid
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationb) (5 points) Give a simple quantum circuit that transforms the state
C/CS/Phy191 Midterm Quiz Solutions October 0, 009 1 (5 points) Short answer questions: a) (5 points) Let f be a function from n bits to 1 bit You have a quantum circuit U f for computing f If you wish
More informationSection 5.5: Matrices and Matrix Operations
Section 5.5 Matrices and Matrix Operations 359 Section 5.5: Matrices and Matrix Operations Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.
More informationSparse-Graph Codes for Quantum Error-Correction
Sparse-Graph Codes for Quantum Error-Correction David J.C. MacKay Cavendish Laboratory, Cambridge, CB3 0HE. mackay@mrao.cam.ac.uk Graeme Mitchison M.R.C. Laboratory of Molecular Biology, Hills Road, Cambridge,
More informationMatrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.
Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n
More informationQuantum Gates, Circuits & Teleportation
Chapter 3 Quantum Gates, Circuits & Teleportation Unitary Operators The third postulate of quantum physics states that the evolution of a quantum system is necessarily unitary. Geometrically, a unitary
More informationQuantum NP - Cont. Classical and Quantum Computation A.Yu Kitaev, A. Shen, M. N. Vyalyi 2002
Quantum NP - Cont. Classical and Quantum Computation A.Yu Kitaev, A. Shen, M. N. Vyalyi 2002 1 QMA - the quantum analog to MA (and NP). Definition 1 QMA. The complexity class QMA is the class of all languages
More information4. Two-level systems. 4.1 Generalities
4. Two-level systems 4.1 Generalities 4. Rotations and angular momentum 4..1 Classical rotations 4.. QM angular momentum as generator of rotations 4..3 Example of Two-Level System: Neutron Interferometry
More informationLecture 20: Bell inequalities and nonlocality
CPSC 59/69: Quantum Computation John Watrous, University of Calgary Lecture 0: Bell inequalities and nonlocality April 4, 006 So far in the course we have considered uses for quantum information in the
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationEXERCISES. a b = a + b l aq b = ab - (a + b) + 2. a b = a + b + 1 n0i) = oii + ii + fi. A. Examples of Rings. C. Ring of 2 x 2 Matrices
/ rings definitions and elementary properties 171 EXERCISES A. Examples of Rings In each of the following, a set A with operations of addition and multiplication is given. Prove that A satisfies all the
More informationCS286.2 Lecture 13: Quantum de Finetti Theorems
CS86. Lecture 13: Quantum de Finetti Theorems Scribe: Thom Bohdanowicz Before stating a quantum de Finetti theorem for density operators, we should define permutation invariance for quantum states. Let
More information6.080/6.089 GITCS May 6-8, Lecture 22/23. α 0 + β 1. α 2 + β 2 = 1
6.080/6.089 GITCS May 6-8, 2008 Lecturer: Scott Aaronson Lecture 22/23 Scribe: Chris Granade 1 Quantum Mechanics 1.1 Quantum states of n qubits If you have an object that can be in two perfectly distinguishable
More information2.0 Basic Elements of a Quantum Information Processor. 2.1 Classical information processing The carrier of information
QSIT09.L03 Page 1 2.0 Basic Elements of a Quantum Information Processor 2.1 Classical information processing 2.1.1 The carrier of information - binary representation of information as bits (Binary digits).
More informationQuantum Mechanics II: Examples
Quantum Mechanics II: Examples Michael A. Nielsen University of Queensland Goals: 1. To apply the principles introduced in the last lecture to some illustrative examples: superdense coding, and quantum
More information6-2 Matrix Multiplication, Inverses and Determinants
Find AB and BA, if possible. 1. A = A = ; A is a 1 2 matrix and B is a 2 2 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find
More informationQuantum Information & Quantum Computation
CS9A, Spring 5: Quantum Information & Quantum Computation Wim van Dam Engineering, Room 59 vandam@cs http://www.cs.ucsb.edu/~vandam/teaching/cs9/ Administrivia Who has the book already? Office hours: Wednesday
More information2 Application of Boolean Algebra Theorems (15 Points - graded for completion only)
CSE140 HW1 Solution (100 Points) 1 Introduction The purpose of this assignment is three-fold. First, it aims to help you practice the application of Boolean Algebra theorems to transform and reduce Boolean
More informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 03/19/2018 Postulates of Quantum Mechanics The postulates (axioms) of quantum mechanics
More informationLOWELL WEEKI.Y JOURINAL
/ $ 8) 2 {!»!» X ( (!!!?! () ~ x 8» x /»!! $?» 8! ) ( ) 8 X x /! / x 9 ( 2 2! z»!!»! ) / x»! ( (»»!» [ ~!! 8 X / Q X x» ( (!»! Q ) X x X!! (? ( ()» 9 X»/ Q ( (X )!» / )! X» x / 6!»! }? ( q ( ) / X! 8 x»
More information1 Quantum Circuits. CS Quantum Complexity theory 1/31/07 Spring 2007 Lecture Class P - Polynomial Time
CS 94- Quantum Complexity theory 1/31/07 Spring 007 Lecture 5 1 Quantum Circuits A quantum circuit implements a unitary operator in a ilbert space, given as primitive a (usually finite) collection of gates
More informationChapter 2. Matrix Arithmetic. Chapter 2
Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the
More informationQuantum Entanglement and the Bell Matrix
Quantum Entanglement and the Bell Matrix Marco Pedicini (Roma Tre University) in collaboration with Anna Chiara Lai and Silvia Rognone (La Sapienza University of Rome) SIMAI2018 - MS27: Discrete Mathematics,
More informationCHAPTER 2 -idempotent matrices
CHAPTER 2 -idempotent matrices A -idempotent matrix is defined and some of its basic characterizations are derived (see [33]) in this chapter. It is shown that if is a -idempotent matrix then it is quadripotent
More informationPh 219b/CS 219b. Exercises Due: Wednesday 22 February 2006
1 Ph 219b/CS 219b Exercises Due: Wednesday 22 February 2006 6.1 Estimating the trace of a unitary matrix Recall that using an oracle that applies the conditional unitary Λ(U), Λ(U): 0 ψ 0 ψ, 1 ψ 1 U ψ
More informationMatrix Dimensions(orders)
Definition of Matrix A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. In general, matrices can contain complex numbers but we
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 1: Quantum circuits and the abelian QFT
Quantum algorithms (CO 78, Winter 008) Prof. Andrew Childs, University of Waterloo LECTURE : Quantum circuits and the abelian QFT This is a course on quantum algorithms. It is intended for graduate students
More informationLecture 2: From Classical to Quantum Model of Computation
CS 880: Quantum Information Processing 9/7/10 Lecture : From Classical to Quantum Model of Computation Instructor: Dieter van Melkebeek Scribe: Tyson Williams Last class we introduced two models for deterministic
More informationLecture 4: Elementary Quantum Algorithms
CS 880: Quantum Information Processing 9/13/010 Lecture 4: Elementary Quantum Algorithms Instructor: Dieter van Melkebeek Scribe: Kenneth Rudinger This lecture introduces several simple quantum algorithms.
More informationDiagonalization by a unitary similarity transformation
Physics 116A Winter 2011 Diagonalization by a unitary similarity transformation In these notes, we will always assume that the vector space V is a complex n-dimensional space 1 Introduction A semi-simple
More informationP = 1 F m(p ) = IP = P I = f(i) = QI = IQ = 1 F m(p ) = Q, so we are done.
Section 1.6: Invertible Matrices One can show (exercise) that the composition of finitely many invertible functions is invertible. As a result, we have the following: Theorem 6.1: Any admissible row operation
More informationCS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games
CS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games Scribe: Zeyu Guo In the first lecture, we saw three equivalent variants of the classical PCP theorems in terms of CSP, proof checking,
More informationQuantum LDPC Codes Derived from Combinatorial Objects and Latin Squares
Codes Derived from Combinatorial Objects and s Salah A. Aly & Latin salah at cs.tamu.edu PhD Candidate Department of Computer Science Texas A&M University November 11, 2007 Motivation for Computers computers
More informationQR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS
QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS DIANNE P. O LEARY AND STEPHEN S. BULLOCK Dedicated to Alan George on the occasion of his 60th birthday Abstract. Any matrix A of dimension m n (m n)
More informationD.5 Quantum error correction
D. QUANTUM ALGORITHMS 157 Figure III.34: E ects of decoherence on a qubit. On the left is a qubit yi that is mostly isoloated from its environment i. Ontheright,aweakinteraction between the qubit and the
More informationTwo matrices of the same size are added by adding their corresponding entries =.
2 Matrix algebra 2.1 Addition and scalar multiplication Two matrices of the same size are added by adding their corresponding entries. For instance, 1 2 3 2 5 6 3 7 9 +. 4 0 9 4 1 3 0 1 6 Addition of two
More informationLecture 9: Phase Estimation
CS 880: Quantum Information Processing 9/3/010 Lecture 9: Phase Estimation Instructor: Dieter van Melkebeek Scribe: Hesam Dashti Last lecture we reviewed the classical setting of the Fourier Transform
More informationBaby's First Diagrammatic Calculus for Quantum Information Processing
Baby's First Diagrammatic Calculus for Quantum Information Processing Vladimir Zamdzhiev Department of Computer Science Tulane University 30 May 2018 1 / 38 Quantum computing ˆ Quantum computing is usually
More informationUnitary t-designs. Artem Kaznatcheev. February 13, McGill University
Unitary t-designs Artem Kaznatcheev McGill University February 13, 2010 Artem Kaznatcheev (McGill University) Unitary t-designs February 13, 2010 0 / 16 Preliminaries Basics of quantum mechanics A particle
More informationLinear Algebra. Alvin Lin. August December 2017
Linear Algebra Alvin Lin August 207 - December 207 Linear Algebra The study of linear algebra is about two basic things. We study vector spaces and structure preserving maps between vector spaces. A vector
More informationBy following steps analogous to those that led to (20.177), one may show (exercise 20.30) that in Feynman s gauge, = 1, the photon propagator is
20.2 Fermionic path integrals 74 factor, which cancels. But if before integrating over all gauge transformations, we shift so that 4 changes to 4 A 0, then the exponential factor is exp[ i 2 R ( A 0 4
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
More information6.896 Quantum Complexity Theory September 18, Lecture 5
6.896 Quantum Complexity Theory September 18, 008 Lecturer: Scott Aaronson Lecture 5 Last time we looked at what s known about quantum computation as it relates to classical complexity classes. Today we
More informationChapter 10. Quantum algorithms
Chapter 10. Quantum algorithms Complex numbers: a quick review Definition: C = { a + b i : a, b R } where i = 1. Polar form of z = a + b i is z = re iθ, where r = z = a 2 + b 2 and θ = tan 1 y x Alternatively,
More informationsparse codes from quantum circuits
sparse codes from quantum circuits arxiv:1411.3334 Dave Bacon Steve Flammia Aram Harrow Jonathan Shi Coogee 23 Jan 2015 QECC [n,k,d] code: encode k logical qubits in n physical qubits and correct errors
More informationStructured Hadamard matrices and quantum information
Structured Hadamard matrices and quantum information Karol Życzkowski in collaboration with Adam G asiorowski, Grzegorz Rajchel (Warsaw) Dardo Goyeneche (Concepcion/ Cracow/ Gdansk) Daniel Alsina, José
More informationD.1 Deutsch-Jozsa algorithm
4 CHAPTER III. QUANTUM COMPUTATION Figure III.: Quantum circuit for Deutsch algorithm. [fig. from Nielsen & Chuang (00)] D Quantum algorithms D. Deutsch-Jozsa algorithm D..a Deutsch s algorithm In this
More informationRobust Quantum Error-Correction. via Convex Optimization
Robust Quantum Error-Correction via Convex Optimization Robert Kosut SC Solutions Systems & Control Division Sunnyvale, CA Alireza Shabani EE Dept. USC Los Angeles, CA Daniel Lidar EE, Chem., & Phys. Depts.
More informationSection Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional)
Section 2.4 Section Summary Sequences. o Examples: Geometric Progression, Arithmetic Progression Recurrence Relations o Example: Fibonacci Sequence Summations Special Integer Sequences (optional) Sequences
More informationQuantum computing and mathematical research. Chi-Kwong Li The College of William and Mary
and mathematical research The College of William and Mary Classical computing Classical computing Hardware - Beads and bars. Classical computing Hardware - Beads and bars. Input - Using finger skill to
More informationQuantum Physics II (8.05) Fall 2002 Assignment 3
Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai
More informationTheoretical Physics II B Quantum Mechanics. Lecture 5
Theoretical Physics II B Quantum Mechanics Lecture 5 Frank Krauss February 11, 2014 Solutions to previous control questions 4.1 Determine first the transformation matrix ˆT through ( ˆT ij = ψ i φ j =
More informationDECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS
DECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS SCOTT HOTTOVY Abstract. Quantum networks are used to transmit and process information by using the phenomena of quantum mechanics.
More informationQuantum Fourier Transforms
Quantum Fourier Transforms Burton Rosenberg November 10, 2003 Fundamental notions First, review and maybe introduce some notation. It s all about functions from G to C. A vector is consider a function
More informationCLASSIFICATION OF COMPLETELY POSITIVE MAPS 1. INTRODUCTION
CLASSIFICATION OF COMPLETELY POSITIVE MAPS STEPHAN HOYER ABSTRACT. We define a completely positive map and classify all completely positive linear maps. We further classify all such maps that are trace-preserving
More informationPh 219b/CS 219b. Exercises Due: Wednesday 20 November 2013
1 h 219b/CS 219b Exercises Due: Wednesday 20 November 2013 3.1 Universal quantum gates I In this exercise and the two that follow, we will establish that several simple sets of gates are universal for
More information