Quantum Computing: Foundations to Frontier

Quantum Computing: Foundations to Frontier CSC2451HF/MAT1751HF Lecture 1

Welcome! Bird s eye view Administrivia 10 minute break Quantum information basics

What you ll get out of this class: Learn the basic principles of quantum information and computation See the principles in action in the most exciting areas of quantum computing Do an in-depth research project into a quantum computing topic you re interested in

Target audience Who this class is for: Computer scientists, physicists, mathematicians, chemists, This is a theory-based course People with a solid linear algebra and probability background Who this class is NOT for: People who want to learn how to build a quantum computer People who want to learn quantum physics

What is quantum computing? It is computation (and more generally, information processing) based on the principles of quantum mechanics, rather than classical physics. Quantum mechanics is a description of nature at its most fundamental level. Formulated in the early 20 th century to explain the behaviour of subatomic particles. QM, and its specializations (quantum field theory, quantum chromodynamics, etc), have been spectacularly successful in explaining microscopic physical phenomena.

The exponentiality of QM After nearly a century of study, the best (classical) methods for predicting the behaviour of general quantum systems require exponential time. The state of N particles requires at least 2 " numbers to describe., = For #~300 (the number of particles in a single uranium atom), 2 " # of atoms in observable universe 10 *+. /. + 1 C +1 Nature seems to be doing extravagant amounts of computation behind the scenes!

The exponentiality of QM So how do physicists actually do (quantum) physics? Answer #1: 100+ years of extremely clever approximations and shortcuts for calculations. Bethe ansatz, Feynman diagrams, perturbation theory, mean field approximations, Answer #2: computer simulations Density functional theory, Quantum Markov Chain Monte Carlo,.. Answer #3: studying systems that allow for Answers #1 and #2 above Systems that are mostly classical Systems with non-strongly interacting particles Lucky that most things we ve studied so far fall in this category! But simulating general quantum physics still seems like a hard task

The exponentiality of QM In 1981, in his talk titled Simulating Physics with Computers, Richard Feynman asked the following question: Can probabilistic computers simulate quantum mechanics? His conclusion, after a lengthy exploration:..quantum mechanics [doesn t] seem to be imitable by a local classical computer. Instead we need a computer built of quantum mechanical elements which obey quantum mechanical laws. Feynman, along with David Deutsch, Paul Benioff and Yuri Manin, are credited with the idea of computing based on quantum mechanical principles.

The early days Feynman s motivation for quantum computers is the most obvious one: simulating quantum systems. (almost tautological!) To physicists in the '80s, this idea might have seemed obvious. To a computer scientist in the 80s, QCs would have seemed like a wild crackpot idea. Also in the 80s, David Deutsch contemplated other applications of quantum computers. Gave an example of a (classical) problem where the parallel universes of QM seem to speed up (by a constant factor) 1992: Dan Simons (UofT PhD!) came up with an example of a problem that could be solved exponentially faster on a QC.

Shor s breakthrough 1993: Peter Shor, inspired by Simons algorithm, realized that the same idea could be extended to give an efficient quantum algorithm for the following problem: (Factorization) Given integer N > 0, find integer M > 1 such that M divides N. The security of the RSA cryptosystem (the most widely deployed public-key encryption on the internet) crucially relies on the assumption that factoring integers is hard!

Crossroads Since Shor s algorithm, physicists and computer scientists have been faced with three options: 1. Quantum mechanics is wrong. 2. There is a fast classical algorithm for factoring. 3. Quantum computers are more powerful than classical computers. At least one of these must be true!

Crossroads Option #1: QM is wrong Perhaps the most successful theory of nature to date. In all its domains of applicability, we have never found experimental disagreement. Option #2: Polynomial-time classical factoring algorithm. RSA has been out for nearly 50 years. There is enormous economic pressure to discover weaknesses, if it exists. The Gauss argument : if the Prince of Mathematics couldn t find it, then it probably doesn t exist.

Crossroads Option #3: Quantum computers are more powerful than classical computers. This would refute the Extended Church-Turing Thesis. Church-Turing Thesis: A Turing machine can simulate any effective computation process. Extended Church-Turing Thesis: A probabilistic Turing machine can efficiently simulate any effective computation process. Option #3 would violate the ECT (assuming factoring is classically hard)!

Present day Major advances on many fronts: Experimental/hardware side Quantum algorithms Connections with fundamental physics and pure mathematics Quantum information theory Quantum complexity theory Quantum cryptography and more!

Emerging quantum computers IBM, Google, Microsoft are big players racing to build (scalable) quantum computers. Also, startups and academic labs are getting into the game Rigetti Computing, Xanadu.ai, IonQ, Dwave, Harvard/U. Maryland/U. Waterloo/U. Bristol/. Different labs/companies are betting on different horses Superconducting qubits, ion traps, photonic systems, topological qubits, Currently in vogue: superconducting qubits

Emerging quantum computers IBM (superconducting qubits) Two 5 qubit and one 16 qubit quantum computers are available on the cloud. Look up IBM Q Experience. You can run programs on their quantum computers using a drag-n-drop interface:

Emerging quantum computers Google (superconducting qubits) Built 9 qubit quantum computer in 2016 Announced a 72 qubit quantum computer (called Bristlecone), but has yet to publish results. Microsoft (topological qubits) Topological quantum computing promises a natural way to implement fault-tolerant qubits and gates. Depends on the existence of non-abelian anyons, which are quasiparticles with exotic topological properties.

Emerging quantum computers Summary There s a qubit race going on, but qubit count is not everything! Current devices are incredibly noisy. Scaling up (i.e. adding qubits) is a tough engineering challenge, but no fundamental obstacles to building a QC with 106 noiseless qubits. Still, large-scale fault-tolerant QCs look like they re many years away. John Preskill: We are in the NISQ era: Noisy Intermediate-Scale Quantum Noisy devices with ~100 qubits. Not capable of running Shor s algorithm, but should still be capable of solving interesting, hard problems. What interesting problems can we solve on near-term quantum computers?

Quantum algorithms Exponential speedups for structured, algebraic problems Factoring (Shor s algorithm) Hidden subgroup problem N = pq Polynomial speedups for unstructured search problems Grover search Exponential speedups for quantum simulation Simulating quantum physics and chemistry (Feynman s dream) Probably the most important application of quantum computers so far!

Quantum algorithms Algorithms to be run on near-term QCs: Variational eigensolvers Classical-quantum hybrid algorithms Quantum Machine Learning Linear system solvers/sdps/convex Optimization Quantum neural nets Recommendation systems more What types of problems admit a quantum advantage?

Connections with fundamental physics The key to a theory of Quantum Gravity could be quantum entanglement, and quantum error correcting codes. The Blackhole Firewall Paradox is an issue about quantum information, and possible resolutions involve quantum cryptography. What can Quantum Computing tell us about Nature?

Uncharted territory

Welcome! Bird s eye view Administrivia 10 minute break Quantum information basics

This class Foundations Basics of quantum information (today, after the break) Quantum circuit model Essential algorithms (Shor, Grover, ) Entanglement Frontier Applications to physics Quantum machine learning Complexity theory Near-term quantum computing Cryptography

Grading 50% Project 40% Problem Sets (~4 psets) I highly suggest partnering up! (no more than groups of 3 please) 10% Participation Lecture scribing (2-3 scribers/lecture) Asking good questions in class Participating in Piazza discussions

Class project Frontier topic of your choosing Project proposal is due October 17 Typical format: read a few research papers, synthesize them into a coherent 10-15 page report and 15 minute presentation Original research encouraged! Other formats: e.g., numerical experiments on IBM s cloud quantum computer Integrate your own research expertise Partner up!

Sign up for Piazza Can get to Piazza through course homepage: www.henryyuen.net/classes/fall2018 Use it to find problem set/project partners Class announcements will be on it Use it to ask questions, or post cool stuff you ve found, or ideas you have!

Resources Textbook: No official textbook, but here are two very good ones: Links to other courses and notes at: www.henryyuen.net/resources

Welcome! Bird s eye view Administrivia 10 minute break Quantum information basics

Bits and qubits Simplest classical system: a bit Two distinguishable states: 0 or 1 Representable as a binary switch, electron spin, photon polarization, transistor voltage level, ON OFF

Bits and qubits Simplest quantum system: a quantum bit (qubit) Mathematically represented as a complex unit vector in C " : # = * + & " + ( " = 1 &, ( C A qubit can represent classical 0 and 1 states: 0 =. / 1 = /. These are orthogonal states Thus, every qubit # is a superposition of the classical 0 and 1 states: # = & 0 + ( 1 &, ( are called amplitudes. They re like probabilities that can be negative! Intuitively: # is in the 0 and 1 state at the same time!

Dirac notation The! notation is called Dirac notation, used to represent quantum states. Mathematically,! ( ket vector ) is a column vector. The dual/hermitian conjugate of column vectors (i.e. row vectors), are called bra vectors :! = -.! = #, & = # 0 + & 1 ket psi bra psi #, & are complex conjugates of #, &. The inner product between two ket vectors! = # 0 + & 1, ( = ) 0 + * 1 is! ( = # ) + & * Notation is helpful for quickly identifying scalars, row and column vectors in complicated expressions. Naming: bra + ket = bracket

Measuring a qubit QM Postulate #1: Quantum states are not directly accessible. They must be measured. If you measure (i.e. look at ) a qubit! = # 0 + & 1, you obtain a classical outcome * {0,1} probabilistically: Outcome 0 with probability # / Outcome 1 with probability & / This is called the Born Rule. If you measure the qubit again, you will obtain * with certainty. In other words, the qubit has collapsed to a classical state; the superposition has been destroyed. Quantum information is fragile!

Measuring a qubit After, w/ prob. * + = +, 1 1 % = 0 % = 2 3 0 + 1 3 1 0 After, w/ prob. - + =., % = 1 Before 0

Evolution of a qubit i.e. not measured. QM Postulate #2: Quantum states in isolation evolve via unitary operations. A unitary operator acting on a qubit is a square matrix! = # $ that takes unit % & vectors to unit vectors: If ' = 0 1 Equivalent properties: is a unit vector, then! ' = 02314 05316 is a unit vector. Preserves inner products: Let ' ( =! ', and ) =! ). Then ' ( ) ( = ' ). The inverse of a unitary! = # $ % & is its conjugate transpose!. = # % $ &.

Evolution of a qubit 1 % = 1 1 ' % = 1 2 0 + 1 2 1 0 0 Before After ' = 1/ 2 1/ 2 1/ 2 1/ 2

A demonstration of quantum weirdness How is a quantum bit different from a probabilistic bit? Experiment A: Let! = 0 and & = ' ( First, apply & to the qubit. Measure the qubit. Apply & again. Measure the qubit. 1 1 1 1. What is the distribution of outcomes?

A demonstration of quantum weirdness Experiment A 1. Apply! to the qubit.! 0 =! % ) = %/ & %/ & = % & 0 + % & 1. " = 0,! = % & 1 1 1 1 2. Measure the qubit: Outcome (a): With probability ½, qubit is 0. Outcome (b): With probability ½, qubit is 1. 3. Apply! to the qubit. (a): w.p. ½,! 0 = % 0 + % 1. & & (b): w.p. ½,! 1 = % & 0 + % & 1. 4. Measure the qubit: No matter what the outcome was in Step 2: Final qubit is 0 with probability ½ and 1 with probability ½. Experiment A behaves like a classical randomization operation.

A demonstration of quantum weirdness Experiment B: Let! = 0 and & = ' ( First, apply & to the qubit. Measure the qubit. Apply & again. Measure the qubit. 1 1 1 1. What is the distribution of outcomes?

A demonstration of quantum weirdness Experiment B 1. Apply! to the qubit.! 0 =! % ) = %/ & %/ & = % & 0 + % & 1. " = 0,! = % & 1 1 1 1 2. Apply! to the qubit.!(! 0 ) =!( % & 0 + % & 1 ) = % & ( % & 0 + % & 1 ) + % & ( % & 0 + % & 1 ) = 1! is a linear operator The zero states cancel out! 4. Measure the qubit: Final qubit is 1 with probability 1! Final outcome is deterministic. This is an example of interference in QM.

A demonstration of quantum weirdness Takeaway: Minus signs in the amplitudes matter! Applying multiple unitary operations to a qubit gives rise to a tree of paths between states. 0 Experiment B Each path has an amplitude associated with it. Before measurement, the paths can constructively or destructively interfere with each other! 0 1 0 1 0 1 In classical probability theory, paths have positive weight that can only add.

A demonstration of quantum weirdness Takeaway: Intermediate measurements destroy superpositions and prevent interference! Measurements make qubits become classical. 0 Experiment A Takeaway: to see quantum effects, delay measurements for as long as possible. This is why it is hard to see quantum effects in everyday life. Measurement is constantly happening. 0 1 0 1 0 1 Intermediate measurements prevent interference.

Composite quantum systems The state of a qubit lives in the vector space C ". Also called the Hilbert space of a qubit. For our (finite-dimensional) purposes, Hilbert space = complex vector space with inner product. The Hilbert space of 2 qubits is the tensor product space C " C " C " has orthonormal basis 0, 1. The tensor product space C " C " C * is 4-dimensional, with orthonormal basis 0 0 = 1 0 0 0 0 1 = 0 1 0 0 1 0 = 0 0 1 0 1 1 = 0 0 0 1 Shorthand: +, = +, = +,. This basis represents the classical states of the two qubits.

Composite quantum systems! ( Tensor product of vectors: if! = # 0 + & 1, and ( = ) 0 + * 1, then the state of the two qubits together is! ( = # 0 + & 1 ) 0 + * 1! ( = #) 0 0 + #* 0 1 + &) 1 0 + &* 1 1 = #) 00 + #* 01 + &) 10 + &* 11 shorthand = #) #* &) &* written explicitly as a vector in C -

Composite quantum systems A two qubit state! is a unit vector in C # C # :! = /,0 1 /0 2 4 /,0 1 /0 # = 1 In general, a two-qubit state cannot be written as a tensor product state! % ' for one-qubit states %, ' C #. States that cannot be written in product form are called entangled. Otherwise, they are unentangled. We will talk much more about entanglement in the next class. Taking inner products in C # C # : let ), *, +,, C # ) * ( +, ) = ) + *,

Composite quantum systems Measuring two-qubit states! = $ %& () C - C - : Obtain classical outcome (, ) 0,1 - - with probability $ %&. The post-measurement state of! is then () Partial measurements: what if we only want to measure the first qubit? - Obtain classical outcome ( 0,1 with probability 2 % = & $ %&. The post-measurement state conditioned on outcome ( is then! % = 1 3 $ 2 %& () = ( 1 3 $ % & 2 %& ) % & Exercise: doing a partial measurement on the first qubit, followed by a partial measurement on the second qubit, yields same distribution of outcomes as doing a full, one-shot measurement.

Composite quantum systems Two-qubit systems in isolation undergo evolution via unitary operators acting on C " C ". Reminder: unitary operators preserve the l " norm/preserve inner products/is inverse to its conjugate transpose. Tensor product of unitaries: Let % = ' (( ' (" ' "( ' "", * = + (( + (" + "( + "" be one-qubit unitaries. Applying % to the left qubit and * to the right qubit, from the perspective of the larger system, corresponds to the unitary (in block matrix form) % * = ' ((* ' (" * ' "( * ' "" * Again, most two-qubit unitaries are not product operators; they are entangling.

The No-Cloning Theorem Classical bits are easily copied. Quantum information is different. Statement: There is no quantum Xerox machine. Formally: there is no unitary! acting on two qubits such that for all one-qubit states ".!( " 0 ) = " " ancilla qubit A proof: Any quantum Xerox machine must also act as a classical Xerox machine. However, classical Xerox machines cannot copy general quantum states.

The No-Cloning Theorem Let! be the two-qubit classical copying unitary that acts as follows: for " 0,1. Equivalently:! " 0 = " "! " 1 = " " 1! = 1 0 0 1 0 1 1 0 This unitary has a special name: it is called CNOT (for controlled-not )

The No-Cloning Theorem * -, 0 = -, - * -, 1 = -, - 1 Let s try to copy! = # $ 0 + 1 using *: *! 0 = 1 2 * 00 + * 10 = 1 2 00 + 11!! This is a very important state known as the EPR pair (or Bell pair, or maximally entangled state). QED.

Composite quantum systems Tensor product spaces (more general) Let! ",! $,,! ' be an orthonormal basis for C ' and * ", * $,, * + be an orthonormal basis for C +. The tensor product space C ' C + is - / dimensional with orthonormal basis! 0 * 1 The inner product on C ' C + is defined as: Warning:! 3 * l (! 0 * 1 ) =! 3! 0 * l * 1 = 9 1 ; = <, l = = 0?. A. When multiplying tensor products of vectors, match up the slots. But not when adding! For example: ( B C ) + ( E F ) ( B + E ) ( C + F )

The exponentiality of QM, redux QM Postulate #3: if the states of systems A and B are! C $, ' C ), respectively, then the joint state of AB is! ' C $ C ). This means that the joint state of + qubits is represented as a vector in C, - C,/ : Each additional qubit doubles the dimensionality of the Hilbert space. Applying a unitary 0 to an +-qubit state! appears to be doing exponentially many computations in parallel:! = 2 3 4,5 / 6 3 7 0! = 2 3 4,5 / 6 3 0 7

The exponentiality of QM, redux Nature is doing an incredible amount of work for us. However, this extravagance is hidden behind the veil of measurement. We can only access the exponential information stored in! in a limited way. This leads to a fundamental tension in quantum information: The exponentiality vs fragility of quantum states This tension makes quantum information and computation very subtle, mysterious, and extremely interesting.

Holevo s Theorem An important mathematical result that captures this tension. Informal statement: one cannot reliably store! bits of classical information in less than! qubits. Formally: Let " <!. Alice gets a uniformly random string $ 0,1 ), and sends an " qubit state * + to Bob, who applies a measurement to obtain a string, 0,1 ). For any encoding/decoding strategy of Alice and Bob, 0 $:, " * + C./ $ 0,1 ), 0,1 )

Next time Quantum entanglement Bell games Certifiable quantum random dice Teleportation Superdense coding Density matrices