beyond breaking rsa: algorithms and applications of quantum computation… · 2020-04-24 ·...

Quantum Theory Quantum Computation Foundations and Complexity Conclusions

”Beyond breaking RSA: Algorithms andapplications of quantum computation”

Carlos Tavares

High-Assurance Software Laboratory/INESC TEC

[email protected]

October 26, 2016


Overview

Quantum Theory

Quantum ComputationQuantum algorithmsHamiltonian simulation

Foundations and Complexity

Conclusions


Introduction


Quantum Theory


Old Quantum Theory

• In the beggining, there were only a set of radical ideas to solveseveral issues in classical physics

• Black-body radiation, Compton and Photoelectric effects,Atomic structure


Foundations of Quantum mechanics

• Find mathematics to quantum mechanics an ever going effort

• ”If quantum physics are waves, were’s the wave equation?”.Erwin Shrödinger formulated the wave equation

• It is a differential equation with potential (manyeigenfunctions and eigenvalues).

What does this mean physically ...?


Quantum mechanics - Strange effects - Superposition

• Eigenvalues and eigenfunctions correspond to observablevalues. Wave function defines probabilities for them - MaxBorn interpretation

• The system may be in several possible states at the same time- superposition principle

Shrödinger’s cat

• Where is the classical limit, e.g. what is a measurement?(Measurement problem)

• Decoherence seems to be the reason that causes wavecollapsing


Quantum mechanics - Strange effects - Entanglement

• Quantum Entanglement• Thought experiment: Einstein-Podolsky-Rosen (EPR) paradox,

or ”spooky action at distance”• Two particles can ”information about a state” a state, e.g.

their local states are highly correlated, regardless of thedistance.

• Experimentally verified by Bell experiments!

• Quantum theory has been extensively verified experimentally,and it is considered as true. It is the most successfull theoryof the history of physics.


Quantum Computation


Quantum Computation

• It took fifty years to discover that quantum effects could helpus store and process information

• Quantum superposition• Qubit is the fundamental unit of information (as similarly to

bit)

|ψ〉 = α |0〉+ β |1〉 ; |α|2 + |β|2 = 1

|0〉 , |1〉 are an orthogonal basis. However the possible statesare provided by all superpositions of the system.


Quantum Computation

• Quantum entanglement• Allows the• The possible states for two qubits is

|ψ〉 = α |00〉+β |01〉+γ |10〉+λ |11〉 ; |α|2+|β|2+|γ|2+|λ|2 = 1

• The information that can be maintained in memory growsexponentially!

• Processing• It is possible to perform operations over all the computational

basis, in a potential exponential time improvement. Suchphenomenon is denominated quantum parallelism.

A |Ψ〉 = A |Ψ1〉+ A |Ψ2〉+ ...+ A |Ψ3〉


Quantum Computation

• No cloning theorem - Information cannot be copied!• x = y , x = x + 1

• Reversible operations - Evolution is unitary UU† = I• Many quantum gates (algebraic operators) are unitary: Toffoli

gates, CNOT, Z Gate

• Semantic observations• Similarities with paralel computation: monoidal categories are

appropriate semantics (they have a tensor product)• Orthogonal bases are classical structures (supply a monoid and

a co-monoid), but states are not

• Decoherence


What can be done withquantum computers?


Quantum computation and simulation - Applications


Breaking RSA cryptography

• The RSA cryptographic system was named after its creatorsRiver, Shamir and Adleman

• Assymetric key encryption system:there is a public and aprivate key

• Two big primes p and q (order ' 10100), and a numbern = p.q. Two numbers d , e, multiplicative inverses of eachother from the totient function of n

• Public key: (n, e); Private key: (p, q, d)• Cyphertext : ce mod n, retrieve original message cd mod n

• The objective is to break the system by finding e and d. Reliesin the hardness of factoring primes.

• It can be broken if one knows the period of the function!


Shor algorithms

• Shor algorithm efficiently calculates the period of a function.• The algorithm operates using two registers of n qubits, one for

the domain and the other for the co-domain• System preparedness

|Φ〉 = 12n/2

(2n−1∑x=0

|x〉

)⊗ |0 . . . 0〉 (1)

• Entangle f (x) in co-domain register with the correspondent xin the domain register

|Ψf 〉 = Uf |Φ〉 =1

2n/2

(2n−1∑x=0

|x ⊗ f (x)〉

)(2)


Shor algorithms• Continuation of the Algorithm:

• We are creating a map between f (x) and the correspondent x .Example for a cyclic function N mod 3

{0, 3, 6, 9, ..., kr} 7→ {0}{1, 4, 7, 10, ..., 1 + kr} 7→ {1}{2, 5, 8, 11, ..., 2 + kr} 7→ {2}

• When the second register is measured the state reads asfollows, where r is what we want to know

|Ψ0〉 =1√K

K−1∑k=0

|x0 + kr〉 (3)

• Apply the Fourier transform and r goes to the amplitude

1

2n/21√K

K−1∑k=0

e2iπy(x0+kr)/2n

|k〉 (4)


Hidden Subgroup

• There is an efficient algorithm to extract r from the state(algorithm must be executed multiple times).

• The algorithm is a specific case of the Hidden SubgroupProblem. It extracts the generator of the hidden subgroup ofa function.

• Can it be generalized to other kinds of groups: non Abeliangroups?

• Some interesting results with no great practical interest.• Desirable: Dihedral group, Symmetrical group;

• Can we do better? Lazlo Babai discovered a sub-exponentialclassical algorithm for the graph isomorfism problem.


Search: The Grover algorithm• Formulated by L.K. Grover in 1996. Applies to the search of

unsorted databases, e.g. it is a general b̈rute forces̈earchalgorithm

• In quantum world: finding the solution is easy, solutionselection is hard. Selection is made by means of a measurent.Build such a measurement is making random pick of a needlein a haystack very probable.


Grover algorithm, Amplitude amplification and estimation

• Magnify the probability of the solution H⊗nXH⊗n

• The application of the oracle application and probabilitymagnification

√(N) times yields the correct solution.

• Grover algorithm is a optimal search algorithm• It is a specific case of a more general technique: Amplitude

amplification, which can be complemented by Amplitudeestimation


Quantum Random Walks

• Quantum walks are analogous to classical random walks.• Three main components: a set of connected nodes, a flipping

coin), an operator that generates the transitions.• A very simple example, a walk in the line:

• Given a state

|Ψin〉 = α↑ |↑〉+ α↓ |↓〉 ⊗ |ψx0〉• The random walk will progress as follows:

U |Ψin〉 = α↑ |↑〉 ⊗ |ψx0−1〉+ α↓ |↓〉 ⊗ |ψx0+l〉• Sample progression


Quantum Random Walks

• The previous example can obviously generalized to moreuseful ones, with an appropriate choice of transition function:trees or graphs

• The technique is useful for problems with a high degree ofbranching. There are some very successfull examples, wherethe benefit is exponential

• It provides an universal model for quantum computation


Hamiltonian simulation


Hamiltonian simulation

• The Hamiltonian of system gives the total energy of a system.The Schrodinger equation defines the probabilistic evolutionof the Hamiltonian in time.

∂ |Ψ〉∂t

= − i~H |Ψ〉

• The system will span the superposition of all eigenstates ofthe Hamiltonian, throughout time.

• This provides an universal model for quantum computing.Establish the initial conditions in the simulation and let timeoperate the system.


Hamiltonian Simulation

• Adiabatic Quantum computation• Provides an universal model for quantum computation.• Specially suited for search and optimization problems

• Continuous Quantum walks• Provides an universal model for quantum computation.• Analogous to discrete quantum walks


Quantum Simulation

Let the computer itself be built of quantum mechanicalelements which obey quantum mechanical laws.(Feynman, 1982).

• Every quantum physical system obeys a dynamics. The idea isto put physics to simulate physics.

• Example: Bose-Einstein condensates simulating black holes,graphene simulating the dirac equation

• Simulators: General or specific, Analog or digital quantumsystems

• Universal computers, condensed-matter physics, atomicphysics, ...;

• Simulated theories• Open quantum systems, chemical models, condensed matter

physics, high-energy physics, atomic physics, cosmology, ....


Foundations for Quantum computation and complexity

• There is an extensive area of work both for the use ofsemantic tools and higher level mathematics in physicaltheories and dynamical systems

• Categorical quantum mechanics, Topos theory, Effectus theory,Logic, extensive use of category theory in geometry andtopology of physical systems

• Objective: conceive tools to the analysis and validation ofquantum algorithms (and their complexity?)

• How to validate hybrid processes (classical and quantum)?• Categorical quantum mechanics, Effectus theory,

generalizations of quantum mechanics, coalgebra theory.



• How to validate processes involving Hamiltonian dynamics ?• Gogioso et al. Monadic dynamics and Categorical semantics

for Shrodinger equation.

• An Hamiltonian simulating a similar Hamiltonian... suggestsan idea of bisimilarity between quantum physical systems...

• Abramsky et al. are already following this line of work


Quantum Complexity

• Complexity is another tool to evaluate computationalprocesses.

• A good survey is available in ”The Complexity Zoo”• All quantum classes are contained in PSPACE


Quantum Complexity classes - BQP

• Bounded Quantum Probability (BQP)• BQP is believed to contain all problems that are solvable in

polynomial time by quantum computers• Many quantum systems have a BQP complete dynamics:

Bose-Hubbard model, XY-Systems.

• Quantum Post Bounded Quantum probability• Problems solvable efficiently by a quantum computer with

post-selection• Could solve all NP-Complete problems, actually it would be

equivalent to the class PP

• NP Complete problems could be solved with non-linearquantum evolution, or with closed timelike curves. Very exoticphysics is necessary!


Quantum Complexity classes - QMA

• Quantum Merlin Arthur• Problems that are really hard to solve even for a quantum

computer. Relatinship with BQP similar to the one between Pand NP.

• Examples of QMA-Complete problems• Quantum k-SAT, k >= 3• Estimate the ground state of K-Local Hamiltonians, for k > 2• Estimate the ground state of Systems of interacting Fermions

or Bosons• Estimate the ground state of Bose Hubbard model

• The many body problem (simulation of a very big number ofparticles) in quantum physics is still beyond reach even forquantum computers.


Conclusions

• There is life beyond RSA in quantum computation.• There are already an interesting set of applications for

quantum computers, and it is still in its infancy!

• Computer science can be very helpful in quantum simulation:help making sense out of myriad of simulations and simulators

• There are still hard problems to quantum computers


References

• Michael A Nielsen and Isaac L Chuang. Quantumcomputation and quantum information. Cambridge universitypress, 2010.

• Complexity Zoo• https://complexityzoo.uwaterloo.ca/Complexity_Zoo

• Quantum algorithm Zoo• http://math.nist.gov/quantum/zoo/

• Stephen Gasiorowicz. Quantum physics. John Wiley & Sons,2007.

• Bob Coecke, New Structures for Physics, Springer, 2010• Carlos Tavares, Foundations for Quantum Algorihms and

Complexity, MAPi pre-thesis, 2015. Available upon request

https://complexityzoo.uwaterloo.ca/Complexity_Zoohttp://math.nist.gov/quantum/zoo/


Questions ?

Quantum TheoryQuantum ComputationQuantum algorithmsHamiltonian simulation

Foundations and ComplexityConclusions

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