Short Course
on
Quantum Computation
The Grand Mathematical Challenge
for the
Twenty-First Century and the Millennium
The American Mathematics Society (AMS) Short Course on Quantum Computation is in conjunction with the AMS Annual Meeting to be held in Washington, DC, January 19-22, 2000.
Synopses and accompanying reading lists follow. Lecture notes will be available to those who register. Advance Short Course registration fees: $80 ($35/student/unemployed/emeritus); on-site Short Course registration fees: $95 ($45 student/unemployed/emeritus). Registration for this Short Course is made through the AMS. One can be registered for the Short Course without being registered for the AMS Meeting.
Please refer to the following websites for more information:
Also at the AMS Annual meeting, an AMS Special Session on Quantum Computation and Information, January 19-21, 2000 has been organized.
Schedule
Monday, January 17, 2000 8:00am - 4:00pm Registration 9:00am - 10:00am Speaker: Samuel J. Lomonaco, Jr. Title: A Rosetta Stone for Quantum Mechanics 10:00am - 10:30am Coffee Break 10:30am - 11:30am Speaker: Peter W. Shor Title: Introduction to Quantum Algorithms 11:30am - 1:30pm LUNCH 1:30pm - 2:30pm Speaker: Howard E. Brandt Title: Qubit Devices 2:30pm - 3:00pm Coffee Break 3:00pm - 4:00pm Speaker: Daniel Gottesman Title: An Introduction to Quantum Error Correction 5:00pm Reception Tuesday, January 18, 2000 9:00am - 10:00am Speaker: Umesh Vazirani Title: Quantum Complexity Theory 10:00am - 10:30am Coffee Break 10:30am - 11:30am Speaker: Alexei Kitaev Title: Quantum Computation with Anyons 11:30am - 1:00pm LUNCH 1:00pm - 2:00pm Speaker: Louis H. Kauffman Title: Quantum Topology and Quantum Computing 2:00pm - 2:30pm Coffee Break 2:30pm - 3:30pm Speaker: Samuel J. Lomonaco, Jr. Title: A Tangled Tale of Quantum Entanglement
The Nobel Laureate Richard Feynman was one of the first individuals to observe that there is an exponential slow down when computers based on classical physics, i.e., classical computers, are used to simulate quantum systems. Richard Feynman then went on to suggest that it would be far better to use computers based on quantum mechanical principles, i.e., quantum computers, to simulate quantum systems. Such quantum computers should be exponentially faster than their classical counterparts.
Interest in quantum computation suddenly exploded when Peter Shor devised an algorithm for quantum computers that could factor integers in polynomial time. The fastest known algorithm for classical computers factors much more slowly, i.e., in superpolynomial time. Shor's algorithm meant that, if quantum computers could be built, then cryptographic systems based on integer factorization, e.g., RSA, could easily be broken. These cryptographic systems are currently extensively used in banking and in many other areas. Lov Grover then went on to create a quantum algorithm that could search data bases faster than any thing possible on a classical computer. These algorithms are based on physical principles not implementable on classical computers, quantum superposition and quantum entanglement.
As a result, the race to build a quantum computer is on. But the mathematical, physical, and engineering challenges to do so are formidable, and are a worthy challenge for the best scientific minds. One of the chief obstacles to creating a quantum computer is quantum decoherence. By this we mean that quantum systems want to wander from their computational paths and quantum entangle with the rest of the environment.
This short course focuses on the mathematical challenges involved in the development of quantum computers and quantum algorithms, challenges worthy of the best mathematical minds. It is hoped that, as a result of this course, many mathematicians will be enticed into working on the grand challenge of quantum computation.
The Short Course will begin with an overview of quantum computation, given in an intuitive and conceptual style. No prior knowledge of quantum mechanics will be assumed.
In particular, the Short Course will begin with an introduction to the strange world of the quantum. Such concepts as quantum superposition, Heisenberg's uncertainty principle, the "collapse" of the wave function, and quantum entanglement (i.e., EPR pairs) will be introduced. This will also be interlaced with an introduction to Dirac notation, Hilbert spaces, unitary transformations, and quantum measurement.
Some of the topics covered in the course will be:
The course will end with a list of the grand challenges of quantum computation, i.e., a list of mathematical problems that must be solved to make the concept of a quantum computer a reality for the twenty-first century and the millennium.
The interaction-free detector [4] provides a simple example of the practical use of path qubits. (The two-dimensional Hilbert space of a path qubit represents two possible quantum-interfering paths of a photon in spacetime.) In this photonic device, the presence of an opaque object in one arm of an interferometer destroys the interference of an incident photon, sometimes signalling the presence of the object, even though the photon could not have taken a path intersecting the object. A simple mathematical analysis of the device is provided.
Another simple example of a photonic qubit device is a quantum key receiver based on a positive operator valued measure [5]. This interferometric device exploits the entanglement of path and polarization qubits.
The mathematical theory of games is currently being generalized to include quantum games. To gain some insight into quantum games, I review a brief mathematical description of a particularly simple quantum game involving quantum-coin flipping [6].
In order to develope a multicomponent qubit device, it is useful to implement various quantum gates. I provide mathematical descriptions of various photonic implementations of quantum gates, including the quantum square-root of not gate, the quantum not gate, the Hadamard gate, and the quantum controlled-not gate. A single-photon balanced Mach-Zehnder interferometer and various photonic qubit entanglers are also described analytically.
Mathematical descriptions are also furnished of a quantum dense coder [7], Bell-state analyzer, entanglement swapper [8], quantum teleporter [9], quantum copier [10], quantum register, quantum factorizer [11], and a quantum error corrector [12].
Brief and critical qualitative descriptions are also given of a full gamut of possible quantum computer implementations, including those based on ion traps [13], cavity quantum electrodynamics [14], nuclear magnetic resonance [15,16], silicon-based nuclear spins [17], quantum dots [18], Josephson junctions [19], superconducting quantum interference devices [20], and neutral atoms [21] and Bose condensates [22] in optical lattices. Finally, quantum robots are briefly reviewed mathematically [23].
D. Gottesman, "Class of quantum error-correcting
codes saturating the quantum Hamming bound," Phys.
Rev. A 54, 1862 (1996).
D. Gottesman, "Stabilizer codes and quantum error
correction," Caltech Ph.D. thesis (1997),
quant-ph/9705052.
P. W. Shor, "Scheme for reducing decoherence in quantum computer memory," Phys. Rev. A 52, 2493 (1995).
L. H. Kauffman and R. A. Baadhio (editors) , Quantum Topology, World
Scientific (1993)
L. H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and
Invariants of Three-Manifolds, Annals of Math. Studies 134, Princeton Univ.
Press (1994).
L. H. Kauffman (editor), Knots and Applications, World Sci. (1995)
L. H. Kauffman (editor), The Interface of Knots and Physics, Proc. of Symp.
in Applied Math. Vol. 51, Amer. Math. Soc. (1996)
L. H. Kauffman, Knots and diagrams, in Lectures at Knots '96, ed. by S.
Suzuki, World Sci. (1997)
D. Aharanov, Quantum Computation, quant-ph/9812037.
Let us consider a graph on a surface of genus $g$. The qubits will be associated with the edges of this graph. One can define operators $A_s=\prod_{j\in {\rm star}(s)}\sigma_x^j$ associated to each vertex. Similarly, there are operators $B_p=\prod_{j\in {\rm boundary}(p)}\sigma_z^j$ associated to the faces. These operators define a quantum code: the codewords are the vectors which satisfy $A_s|\xi\rangle=|\xi\rangle$ and $B_p|\xi\rangle=|\xi\rangle$ [1]. Such vectors form a subspace of dimensionality $2^{2g}$. Changing one of the stabilizer conditions to $A_s|\xi\rangle=-|\xi\rangle$ or $B_p|\xi\rangle=-|\xi\rangle$ can be interpreted as an ``excitation''. The excitations reveal nontrivial properties even on the plane: if one moves one excitation around the other, the quantum state is multiplied by $-1$. Such multiplication by a phase factor is a characteristic feature of Abelian anyons.
A generalized version of this model gives rise to nonabelian anyons. Each qubit can be replaced by a larger quantum system whose basis states are indexed by elements of any finite group $G$. From the physical point of view, the new model is an implementation of a discrete gauge symmetry [2]. One can associate a finite-dimensional Hilbert space to each excitation configuration on the plane. It is interesting that this space does not have a tensor product structure. More exactly, it has a tensor factor associated with each excitation, but these tensor factors are not protected against errors. The remaining factor, which is protected, is non-local (i.e. depends on all the excitations together). One can act on the protected Hilbert space by moving excitations around each other. Each braid group element (i.e. a topologically different way of moving the excitations) is represented by a certain unitary operator. If one fuses two excitations into one, the Hilbert space shrinks. Actually, it splits into several Hilbert spaces corresponding to different types of the new excitation. Thus fusing two excitations is a measurement. Finilly, if one creates a new pair of excitations, it always appears in a certain quantum state. All three operations, braiding, fusion, and creation of a new pair, are intrinsically fault-tolerant due to their topological nature [3].
An important question about anyons is whether the topological operations form a universal computational basis. This turns out to be the case for $G=S_3$, despite the fact that the image of the braid group in the group of unitary operators is finite (for any given number of strands). Universality is achieved in an adaptive manner, i.e. by doing measurements during computation and choosing the next braid group generator depending on the previous measurement outcomes.
It should be noted that anyons are not necessarily related to groups. The most general mathematical framework for anyons is a unitary ribbon category. This type of an algebraic structure has been studied in connection with braid group representations and invariants of knots and 3-manifolds [4].
2. F.A.Bais, P. van Driel and M. de Wild Propitius, Quantum
symmetries in discrete gauge theories, Phys.~Lett. B280, 63 (1992).
3. A.Yu.Kitaev, Fault-tolerant quantum computation by anyons,
http://xxx.lanl.gov, quant-ph/9707021 (1997).
4. Turaev, V. G. Quantum invariants of knots and 3-manifolds,
de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co.,
Berlin, 1994.
We begin by noting that the state | \Psi > of a quantum system can be represented as either an element of a Hilbert space H (usually finite dimensional for quantum computing devices), or as a positive definite operator \rho (having trace 1) on H, called the density operator. We then consider quantum systems "living" respectively in the Hilbert spaces H_1, ... , H_n. They are said to be entangled if their joint state | \Psi >, which "lives" in the tensor product H of H_1, ..., H_n, cannot be factored into the tensor product of states | \Psi_j > "living" respectively in H_j, j = 1 ... n. There is a similar definition in terms of density operators.
After the definition of QE, we discuss the Einstein-Podolsky-Rosen (EPR) paradox. Then we give two examples of applications of QE, namely, 1) quantum teleportation and 2) the Shor quantum factoring algorithm. In each of the two examples, we draw a contrast between two perspectives, i.e., the Schrodinger and the Heisenberg models of quantum mechanics.
Next we proceed to study the mathematical structure of QE. Let SU_j (j=1,2, ... , n) and SU denote respectively the Lie groups of special unitary transformations on H_j and H. We define the locality subgroup L(n) of SU as the tensor product of the n Lie groups SU_j. Two quantum states \rho and \rho' are said to be locally equivalent if there is a unitary transformation in L(n) which transforms \rho into \rho'. Under this equivalence relation, the Lie algebra u of the unitary group U of H decomposes into L(n)-orbits, called entanglement classes. The entanglement classes turn out to be symplectic manifolds with a rich mathematical structure. We will construct invariants of QE.
Finally, we explore the intriguing possibility of a relationship between QE and knot theory. We will, for example, illustrate a relationship between the GHZ state and the Borromean rings of knot theory.
If time permits, we will give a list of mathematical research problems.
[ 2] Bennett, Charles H., David P. DiVincenzo, Christopher A.
Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and
William K. Wootters, Quantum nonlocality without
entanglement, quant-ph/9804053.
[ 3] Berman, Gennady, Gary D. Doolen, Ronnie Mainieri, and
Vladimir I. Tsifrinovich, "Introduction to Quantum
Computation," World Scientific (1999).
[ 4] Deutsch, David, and Patrick Hayden, Information flow in
entangled quantum systems, quant-ph/9906007.
[ 5] Freedman, Michael H., and David A. Meyer, Projective
plane and planar quantum codes, quant-ph/9810055.
[ 6] Freedman, Logic, P/NP, and the quantum field computer,
Proc. Natl. Acad. Sci. 95 (1998), 98-101.
[ 7] Freedman, Michael H., \Topological views on
computational complexity, preprint.
[ 8] Gottesman, Daniel, The Heisenberg representation of
quantum computers, quant-ph/9807006.
[ 9] Kauffman, Louis H., "Knots and Physics," World Scientific
Publishers (1991), Second Edition 199.
[10] Kitaev, A. Yu, Quantum computation: Algorithms and
error correction, Uspekhi Mat. Nauk 52 (1997), pp 53-112.
[11] Linden, N., and S. Popescu, On multi-particle
entanglement, quant-ph/9711016.
[12] Linden, N., S. Popescu, and A. Sudbery, Non-local
properties of multiparticle density matrices, quant-ph/9801076.
[13] Lo, Hoi-Kwong, Tim Spiller & Sandu Popescu (editors),
"Introduction to Quantum Computation & Information," World
Scientific (1998).
[14] Lomonaco, Samuel J., Jr., A quick glance at quantum
cryptography}, Cryptologia, Vol. 23, No.1, January,1999,
pp1-41. (quant-Ph/9811056)
[15] Lomonaco, Samuel J., Jr., A rossetta stone for quantum
mechanics, (to be enclosed with course materials), in
preparation.
[16] Milburn, Gerald J., "The Feynman Processor," Perseus
Books, Reading, Massachusetts (1998).
[17] Sakurai, J.J., "Modern Quantum Mechanics," Addison-Wesley,
(1994). (Revised edition)
[18] Williams, Collin P., and Scott H. Clearwater, "Explorations
in Quantum Computation," Springer-Verlag (1997).
[19] Wootters, William K., Entanglement of formation of an
arbitrary state of two qubits, quant-ph/9709029.
While based on quantum mechanics, the mathematical model for quantum computation is straightforward and can be understood without a deep physics background. The state of the quantum computer is an element of a tensor product of two-dimensional complex vector spaces, each of which we call a qubit. The computation is performed by manipulating the state of the computer via a sequence of unitary transformations; in each of these transformations, at most two of the qubits generating the tensor product are allowed to interact. (Replacing two by any larger, constant, integer generates the same model of computation.) To output the result, a von Neumann projection measurement is used to extract information from the state of the computer.
As mentioned above, to date only a very few classes of algorithms have been discovered where quantum computers provide a substantial speedup. One class is based on using Fourier transforms to find periodicity. The first such algorithm was discovered by Dan Simon, and the algorithms for factoring integers and computing discrete logarithms that I discovered also fall into this class. A second class includes Lov Grover's algorithm, which speeds up the time for searching for an item in an unordered list from linear in N to order of the square root of N, where N is the number of items in the list. A number of interesting extensions of this algorithm have been discovered, all using essentially the same techniques. The third class consists of algorithms for simulating quantum mechanical systems. I plan to explain the basic model of quantum computation, and show how the first two classes of algorithms described above work.
Since quantum computation is on the whole a very young field, relatively few overviews of the area have been written. Two good ones are Alexei Kitaev's survey article [4] and John Preskill's lecture notes [5]. The other references listed below are an article proving some basic theorems about the model of quantum computation [1], and four papers explaining some of the algorithms described above.
(2) L. K. Grover, "Quantum mechanics helps in searching for a needle
in a haystack," Phys. Rev. Letters, 78, pp. 325-328 (1997).
(3) L. K. Grover, "A framework for fast quantum mechanical algorithms,"
in Proceedings of the Thirtieth Annual ACM Symposium on Theory of
Computing, pp. 53-62, ACM Press, New York (1998).
(4) A. Yu. Kitaev, "Quantum computation, algorithms, and error correction,"
Russian Math. Surveys, 52:6, pp. 1191-1249 (1997).
(5) J. Preskill, Lecture Notes for Physics 229, Caltech, available on-line
at http://www.theory.caltech.edu/people/preskill/ph229/.
(6) P. W. Shor "Polynomial-time algorithms for prime factorization and
discrete logarithms on a quantum computer," SIAM J. Computing, 26,
pp. 1484-1509 (1997).
(7) D. R. Simon "On the power of quantum computation," SIAM J. Computing,
26, pp. 1474-1483 (1997).
The complexity class that captures the power of quantum computers is BQP - the class of languages that can be recognized (with bounded error probability) in polynomial time on a quantum Turing Machine. We know that BQP contains the class BPP of languages that can be recognized in polynomial time on a probabilistic Turing Machine, and is in turn contained in the class $P^{\#P}$, of counting problems. One of the most important open questions in this area is: does BQP contain NP? An affirmative answer would mean that quantum computers can efficiently solve many of the most important computational problems, including the traveling salesman problem. There is evidence showing that this will be a hard question to resolve - in much the same way as the P vs. NP problem. It has been proved that in the black box model, that quantum computers cannot solve NP-complete problems in subexponential time [4] (indeed even in o(2^{n/2}) steps).
A study of the relationship between quantum computation, nondeterminism and interaction (as computational resources) has already proved to be quite fruitful. It has been recently shown that BQNP - the quantum analog of the class NP - is contained in $P^{\#P}$ [5]. There have also been significant developments in the area of quantum interactive proofs. Here the issue is the number of rounds of communication and number of quantum bits that must be exchanged between a prover and verifier, to convince the (polynomially bounded) verifier of the answer to a computational problem. It has been shown that PSPACE has a two round quantum interactive proof [6].
Quantum communication complexity is another area that has been extensively studied in the last few years, and have provided insights into quantum computation and vice-versa. The power of quantum computation lies in the exponentially many hidden degrees of freedom in the state of an $n$ quantum bit system --- whereas $2^n - 1$ complex numbers are necessary to specify the state, Holevo's theorem states that $n$ quantum bits cannot be used to communicate any more than $n$ classical bits. Nevertheless, it has recently been established that there are communication tasks that can be carried out using exponentially fewer quantum bits than classical bits [7][8].
In this talk I will give an overview of these new quantum complexity classes, some of the techniques used in proving the main results in the area, and a discussion of the open issues in the area. A set of lecture notes covering some of these topics can be found at the web site [9].
[1] E. Bernstein, U. Vazirani, "Quantum complexity theory", {\em Siam Journal of Computing}, {\bf 26}, October, 1997 (special issue on quantum computation). Revision of {\em Proc. 25th Annual ACM Symposium on Theory of Computing}, 1993, pp. 11-20.
[2] P. Shor, "Algorithms for quantum computation: Discrete logarithms and factoring," Siam Journal of Computing, 26, October, 1997, pp. 1484-1509 (special issue on quantum computation). Revision of Proc. 35th Symposium on Foundations of Computer Science (FOCS), 1994.
[3] D. Simon, "On the power of quantum computation." In Proc. 35th Symposium on Foundations of Computer Science (FOCS), 1994.
[4] C. Bennett, E. Bernstein, G. Brassard, U. Vazirani, "Strengths and Weaknesses of Quantum Computation," Siam Journal of Computing, 26, October, 1997, (special issue on quantum computation).
[5] A. Kitaev, "Quantum NP," AQIP conference, Chicago, January 1999.
[6] J. Watrous, "PSPACE has a 2 round quantum interactive proof," to appear Proceedings of Symposium on the Foundations of Computer Science, 1999.
[7] A. Ambainis, L. Schulman, A. Ta-Shma, U. Vazirani, A. Wigderson, "The Quantum Communication Complexity of Sampling," Proceedings of Symposium on the Foundations of Computer Science, 1998.
[8] R. Raz, Proc. 31st Annual ACM Symposium on Theory of Computing, 1999.
[9] U. Vazirani, "Course Notes on Quantum Computation", http://www.cs.berkeley.edu/~vazirani/qc.html
Last update: November 11, 1999