A mathematical view of quantum computation J UANJO R U AND S EBASTIAN - - PowerPoint PPT Presentation
Q UANTUM C OMPUTATION AND I NFORMATION S EMINAR S ECURITY AND Q UANTUM I NFORMATION G ROUP T ECHNICAL U NIVERSITY OF L ISBON 9 October 2009 A mathematical view of quantum computation J UANJO R U AND S EBASTIAN X AMB F ACULTAT DE M
QUANTUM COMPUTATION AND INFORMATION SEMINAR SECURITY AND QUANTUM INFORMATION GROUP TECHNICAL UNIVERSITY OF LISBON
9 October 2009
JUANJO RUÉ AND SEBASTIAN XAMBÓ FACULTAT DE MATEMÀTIQUES I ESTADÍSTICA UNIVERSITAT POLITÈCNICA DE CATALUNYA 08028 BARCELONA (SPAIN)
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JOINT WORK WITH JUANJO RUÉ Quantum Computation: Foundations and State of the Art (Master Thesis UPC)
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MAIN POINTS
Introduction Quantum computation in mathematical terms
Comments about some q‐algorithms Quantum computation in physical terms
Ending remarks
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ABSTRACT
A mathematical model of a quantum computer, or q‐ computer, will be presented, together with related concepts such as q‐gates, q‐computations and q‐algorithms/programs. Emphasis will be given to examples, such as the q‐Fourier transform and q‐algorithm of Shor to factor integers in poly‐ nomial time. The possible physical realizations of the model will be analyzed using an axiomatic version of quantum me‐
tioned.
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INTRODUCTION
COMPUTACIÓN Level CLASSICAL QUANTUM Mathematical Mathematical logic
Turing machine
Boole algebra (Shannon)
von Neumann machines Algorithmic theory Parallel computing
Linear algebra Vectors
Matrices
Physical theory Mechanics Electromagnetism Quantum Mechanics (basic axioms) Technology Circuits, transistors, … Ionic traps, … Economics Ubiquity of processors computers mobile phones digital cameras, … Future
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QUANTUM COMPUTATION IN MATHEMATICAL TERMS
Notations positive integer (number of bits or q‐bits) positive integer in the range 0 ∙∙ 2 1 binary expression of ( 2 2) space of ‐vectors of order : ∑
These are complex vectors of 2 components:
1
1
∑
|
(we say that |0, |1, …, |2 1 is an orthonormal basis)
If , we write (conjugate of )
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Example ( 1) |0 |1 1 0 0 1. Example ( 2) |0 |1 |2 |3 |00 |01 |10 |11
1 1 1
In general, | | | | | ||
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Proof |0 |0 1 0 1
1
|00 |0 |1 1 0 0 1
1
|01 |1 |0 0 1 1
1
|10 |1 |1 0 1 0 1
1
|11
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q‐Computation
If is a matrix, its transpose is and its adjoint
A ‐computation of order n is a matrix , , , such that (that is, is a unitary matrix of order 2 1: 2 ). If , , and . In other words,
computation of order ; and
computation of order .
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A q‐input for a q‐computation is a vector such that | 1 (unitary vector). Example: |0 |1 |2 1/√2 The q‐output of a q‐computation is the (unitary) vector . Examples ( 1). A q‐computation of order 1 is a matrix , i.e., a matrix of the form
1.
any unitary matrix of order 2 has this form.
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Special cases: a) Pauli matrices
1 0 0 1, 0 1 1 0, 0
1 |0 |1, |1 |0
b) Hadamard matrix
√ 1
1 1 1 |0 |0 |1/√2 |1 |0 |1/√2
c) Fase matrices
1 / / / In particular, / 1 and / 1 /
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Examples 2 Let . The we define as follows: |0 |0, |1 |1|. If , then 1 0 0 0 0 1 0 0 0 0 0 0
|0 |0, |1 |1|1 : 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
it according to whether the first bit is 0 or 1. It is a conditional negation (CONTROLED‐NOT)
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is defined in an analogous way. For example, 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
1 0 0 0 0 0 0 0 0 1 0 0 0 0 1
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q‐Computer
A q‐computer of order is a system that allows to perform the following
1. Selects the unitary vector .
|
| |
3.
,
| 1 0 | 1 1 | 1 1 | 1 0
Returns 0 2 1 with probability || and resets as |. Negationof the k‐th bit if the j‐th bit is |1 (see examples , and , above for 2) Initialization or Input Action on the j‐th bit Observation of
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A q‐algorithm is a sequence , … , such that each is either
,, and we say that it performs the q‐computation
( is called the complexity
Example Swap (trasposition of 2 bits) SWAP,
,
Indeed,
|
,: |
|
|
|
SWAP[0,1] 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1
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Theorem Any q‐computation can be realized by a q‐algorithm. A q‐algorithm is said to be special or restricted if the operations , ap‐ peraring in it are such that is one of the following three matrices:
1 1 1 , / 1 , / 1 / . Theorem For any q‐computation there exists a special q‐algorithm that performs the computation with as much approximation as wanted. The basic idea of the proof is that any can be approximated to any wanted degree by products formed with matrices taken from , /, /.
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Example The discrete Fourier transform of is the linear operator : , |
where / /. Observe that : | | |
0 if , for, if 0, ∑
Let us give an idea of how to produce a ‐algorithm to obtain .
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After some calculations we get that |
0 // 1
√ | √ | √ |,
where ,/ . This shows that can be computed by a ‐algorithm of complexity 1 1 1 2
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q‐Programs
A [restricted] q‐program has the following structure: INITILAZATION
, … , OUTPUT The vector OBSERVATION [Optional]
number (the complexity of the algorithm). We say that an algorithm is polynomial if its complexity is bounded by a polynomial in .
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Example Generator of random numbers in the range 0 2 1 with uniform distribution: RANDOM
1/2.
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COMMENTS ABOUT SOME POLYNOMIAL Q‐ALGORITHMS
‐Fourier transform. Estimate of the phase of the eigenvalue of a unitary operator given the eigenvector | and the operators
.
Given positive integers and , , such that gcd, 1, to
find the least positive integer such that 1 mod (Shor)
Given a positive integer, to find its factorization (Shor) (Discrete logarithm): Given positive integers , and (, ) to
find the least positive integer such that mod , if it exists (Shor).
(Grover’s algorithm). To finding an arbitrary element in a database of
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QUANTUM COMPUTATION IN PHYSICAL TERMS
dowed with a Hermitian scalar product | (i.e., linear in and linear‐ conjugate in ). For the purposes of quantum computation we may also asume that has finite dimension. The non‐zero vectores represent states of Σ, and two non‐zero vectors , represent the same state if and only if there exists such that . In particular, any state can be represented by a unitary vector (deter‐ mined up to a phase factor ). Thus the state space of Σ is (the pro‐ jective space associated to ). Following Dirac, we will write | to denote the state corresponding to (in projective geometry it is denoted ). Quantum superposition: | | .
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| | . If we express with respect to an orthonormal basis, it is easy to check that this condition is equivalent to
If , … , are the distinct eigenvalues of , then , … , and ∑
: is the orthogonal projection onto the
space | of eigenvectors of with eigenvalue . The result of an observation or measure of when Σ is in the state | is
Σ is reset to the state
(or, more precisely, to | ).
In particular, if , then the observation yields with certainty and Σ remains in the state |.
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Example (Eventualities). If is a subspace of , the orthogonal projection : is an observable with eigenvalues 1 and 0: , . The observables of this form are called eventualities.
vironment is not affected by Σ) in the time interval 0, , there exists a unitary operator : such that represents the state of Σ at time if represents the state of Σ at time 0.
, then the associated space of the composite system Σ Σ is .
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q‐bits (qubits)
The states of a spin ½ particle (system Σ) can be thought as points lying
The complex space associated to this system according to axiom 1 is (spinor space). This fact can be argued as follows.
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point
, , | 1
Setting
, we get a bijection between
∞ and . The inverse map is given by , ,
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, for any element
, is proportional to a unique vector of the form 1, when 0, and to 0,1 if 0. Thus we have a map
The inverse map is given by , / if 0 ∞ if 0 Therefore
.
This shows that the space associated to Σ is .
, is called the Riemann
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q‐Registers
By axiom 4 and the formula , the space is the associated space of Σ Σ Σ ( summands), the sys‐ tem composed of q‐bits (and which is called a q‐register of order ). Then axiom 3 tells us that the time evolution of Σ is given by a unitary matrix of order 2. In other words, the time evolution of Σ is a q‐ computation. Finally, axiom 2 indicates that the [optional] operation at the end of q‐programs corresponds to the operation of measuring the (diagonal) ob‐ servable ∑ ||
when Σ is in the state |. Note that |, hence
| and
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Example (EPR states). A possible state of a ‐register of order 2 is |
√ |00 |11.
Such states are called EPR states (also called entangled states). Suppose the first ‐bit is located at and the other at . Then if an ob‐ server at measures the first ‐bit and then and observer at measures the second ‐bit, we note that they get the same value: State 0 |00 0 1 |11 1 Indeed, the EPR state | “collapses” to |00 or |11 if measures 0 or 1, respectively (ie, the projection of | to the space |0 is |00, and to the space |1 is |11).
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QUANTUM COMPUTERS
From the preceding observations it follows that it is sufficient, in order to execute q‐programs of order on a physical support, to have a quantum register Σ and “implementations” of the operations ,
A quantum computer (of order ) is a quantum register Σ endowed with such implementations. Its main beauty is that such a computer allows us to perform (or approx‐ imate) any ‐computation.
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ENDING REMARKS
Quantum parallelism This feature stems from the possibility of initializing the ‐computation in states such as |0 |1 |2 1/√2:
This state contains (actually is the normalized sum of) all numbers of
bits.
Hence, any operation of the quantum computer acts on all numbers
much faster than a classical computer.
In general, the usefulness of the algorithms (as Shor’s factorization,
for instance) is based in the fact that after its execution the ampli‐ tudes of “useful numbers” are high and the others are small.
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The problem of decoherence This difficulty arises from the fact that interactions with the environment can quickly “perturb” the states of Σ (uncontrolled “entanglement” be‐ tween states of the environment and of Σ). Such problems in the road of building quantum computers are of a “phys‐ ical” nature. Research in many labs around the world is focussed on those questions, with continuous progress and in many direction: http://en.wikipedia.org/wiki/Timeline_of_quantum_computing (we see an explosion of activity in the last years, and especially since 2006). See http://en.wikipedia.org/wiki/Quantum_computer for over a dozen lines of inquiry toward the realization of a quantum computer. (On 24 February 2009 D‐Wave Systems announced a 128 q‐bits QC)
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Teleportation The techniques of quantum computation allow to transfer the state of q‐ bit at to the same state of a ‐bit at (the state disappears at and appears at ). Here is a sketch of the procedure.
ported from to .
||
And collecting with respect to the first two ‐bits,
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|10 |0 |1 |11 |1 |0
state of the ‐bit at are given in the following table: Value 00 01 10 11 State |0 |1 |1 |0 |0 |1 |1 |0
sical bits produced by the measures of by applying the gates , , , , respectively. Recently this possibility has been demostrated with Yb atoms at a dis‐ tance of 1m (Olmschenk et al. 2009). This opens great potential for quantum networks.
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REFERENCES
Mermin, N. David [“known for the clarity and wit of his scientific writings”] Quantum Somputer Science: an introduction Cambridge University Press, 2007 Kaye, Phillip ‐ Laflamme, Raymond ‐ Mosca, Michele An introduction to quantum computing Oxford University Press, 2007 Jaeger, Gregg Quantum Information―An overview Springer, 2007 Parthasarathy, K. R. Lecture notes on quantum computation, quantum error correcting codes and in‐ formation theory Tata Institute of Fundamental Research, Narosa Publishing House, 2006 (distributed by the AMS)
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Quantum computation and quantum information Cambridge University Press, 2000 Stephen P. Jordan Quantum Computation Beyond the Circuit Model Tesis MIT, Sep. 2008 Peter W. Shor. Polynomial‐time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484‐ 1509, 2005.
nroe: Quantum teleportation between distant matter Qbits. Science, 323 (23 Jan 2009). Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6/7):467‐488, 1982
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