Introduction To Quantum Computing Day 2 - Quantum Communication And .
Introduction to Quantum ComputingDay 2 - Quantum Communication and CryptographyMengoni Riccardo, PhD24 June 2021
Content Recap of QM Quantum Communication Quantum Teleportation Superdense Coding Quantum Cryptography Quantum Key Distribution
Recap of QM
Linear AlgebraTensor ProductCompact form:Dimension 𝑛2
Postulates of Quantum Computing (1)QuantumlyTo a closed quantum system is associated a space of states H which is aHilbert space. The pure state of the system is then represented by aunit norm vector on such Hilbert space.The unit of quantum information is the quantum bit a.k.a. QubitState of a qubit:
Postulates of Quantum Computing (1)Space of states:State of a qubit:Canonical basis:Other basis:
Postulates of Quantum Computing (1)Different physical realization of qubitsCanonical Basis
Postulates of Quantum Computing (2)QuantumlyThe space of states of a composite system is the tensorproduct of the spaces of the subsystemsState of N qubits:
Postulates of Quantum Computing (2)Quantum EntanglementStates that can NOT be written as tensor product are entangledBell’s states
Postulates of Quantum Computing (3)QuantumlyThe state change of a closed quantum system isdescribed by a unitary operatorSchrodinger Equation
Postulates of Quantum Computing (4)Quantumly To any observable physical quantity is associated an hermitian operator OO 𝑜𝑖 ۧ 𝑜𝑖 𝑜𝑖 ۧ A measurement outcomes are the possibile eigenvalues 𝑜𝑖 .The probability of obtaining 𝑜𝑖 as a result of the measurement isPr(𝑜𝑖 ) 𝜓 𝑜𝑖 2The effect of the measure is to change the state 𝜓ۧ into the eigenvector of O 𝜓ۧ 𝑜𝑖 ۧ
Postulates of Quantum Computing (4)Different physical realization of qubitsObservable quantitiesO 𝑜𝑖 ۧ 𝑜𝑖 𝑜𝑖 ۧCanonical Basis
Postulates of Quantum Computing (4)Different physical realization of qubitsObservable quantities𝒁 0ۧ 0ۧ𝒁 1ۧ 1ۧ𝑿 ۧ ۧ𝑿 ۧ ۧCanonical Basis
Postulates of Quantum Computing (4)Different physical realization of qubitsObservable quantitiesCircularLinearPolarization Polarization𝑿𝒁Canonical Basis
Postulates of Quantum Computing (4)OutcomeExample:MeasurementInitial qubit state𝒁𝑿 0ۧcon 1ۧcon ۧcon ۧcon
Quantum Communication
Quantum CommunicationClassical vs Quantum ChannelClassical information channel is a communication channel used totransmit classical informationUnit of classical information - BIT 𝟎, 𝟏Example: transmissioncables (channel) ofelectrical impulses(classical information)
Quantum CommunicationClassical vs Quantum ChannelQuantum information channel is a communication channel that can be usedto transmit quantum informationUnit of quantum information - QUBITIt is capable of transmitting not only base states ( 0ۧ, 1ۧ) but also theirquantum superimpositions (e.g. 0ۧ 1ۧ ).Coherence is maintained while transmitting through the channel.
Quantum CommunicationQuntum information encoded into photonsQuantum Channel:Free-space
Quantum CommunicationQuntum information encoded into photonsQuantum Channel:Optical fiber
Quantum CommunicationThe realization of quantumcommunication protocols isalready possible today thanks tospecialized devices capable ofmeasuring qubits. Therefore, theimplementation of these protocolsQuantum Channel:does not necessarily require theOptical fiberpresence of a quantum computer
Quantum CommunicationNo-Cloning TheoremGiven the postulates of quantum mechanics, it not possible to copyexactly (cloning) an unknown quantum stateDoes not exist an operator U such that, given a staterelizesOn the other hand, it is possible to perform cloning if the state belongs toan orthogonal set of states - e.g. when it is a classic state
Quantum Teleportation
Quantum Teleportation
Quantum Teleportation
Quantum TeleportationAlice wants to send a quantum state to Bob, having only a classicalcommunication channel available.Specifically, suppose you want to send the state of the qubit labelled C.Classical communication channel
Quantum TeleportationRemember that Alice cannot make a copy of the state of her qubit dueto the No-Cloning theoremClassical communication channelCloning.
Quantum TeleportationAlice and Bob share a pair of entangled qubits (named A and B)transmitted to them by an Entangled Qubit source (via quantum channels)Classical communication channelBAEQS
Quantum TeleportationThe global state of the three qubits possessed by Alice and Bob is
Quantum TeleportationThe global state of the three qubits possessed by Alice and Bob isThe state of qubit Cthat Alice wants tosend to BobEntangled qubit pair A and Bshared by Alice and Bob
Quantum TeleportationThe global state of the three qubits possessed by Alice and Bob isUsing the following relation (Bell states)
Quantum TeleportationIt is possible to rewrite the global state asTeleportation occurs when Alice measures hertwo qubits A and C in the Bell basis
Quantum TeleportationThe result of Alice's measurement is that the state of the threequbits collapses into one of the following four states (with equalprobability). Alice uses two-bit encoding (also known to Bob) todescribe the measurement resultEncoding00011011
Quantum TeleportationThe result of Alice's measurement is that the state of the threequbits collapses into one of the following four states (with equalprobability). Alice uses two-bit encoding (also known to Bob) todescribe the measurement resultEncoding00011011
Quantum TeleportationAlice sends the two bits of information to Bob via the classic channel.Bob applies appropriate local operations to achieve teleported stateClassical communication channel00
Quantum TeleportationThe result of Alice's measurement is that the state of the threequbits collapses into one of the following four states (with equalprobability). Alice uses two-bit encoding (also known to Bob) todescribe the measurement resultEncoding00011011
Quantum TeleportationAlice sends the two bits of information to Bob via the classic channel.Bob applies appropriate local operations to achieve teleported stateClassical communication channel01Z Bob applies theZ operator
Quantum TeleportationThe result of Alice's measurement is that the state of the threequbits collapses into one of the following four states (with equalprobability). Alice uses two-bit encoding (also known to Bob) todescribe the measurement resultEncoding00011011
Quantum TeleportationAlice sends the two bits of information to Bob via the classic channel.Bob applies appropriate local operations to achieve teleported stateClassical communication channel10X Bob applies theX operator
Quantum TeleportationThe result of Alice's measurement is that the state of the threequbits collapses into one of the following four states (with equalprobability). Alice uses two-bit encoding (also known to Bob) todescribe the measurement resultEncoding00011011
Quantum TeleportationAlice sends the two bits of information to Bob via the classic channel.Bob applies appropriate local operations to achieve teleported stateClassical communication channel11ZX Bob applies theZX operator
Quantum TeleportationQuantum Teleportation Protocol: Final Comments Quantum teleportation is not instantaneous: in order to reconstructthe initial state, Bob must first receive the two bits associated withAlice's measurement. These are transmitted via a classicalcommunication channel, so the signal cannot travel at superluminalspeed (in accordance with special relativity). Quantum teleportation respects No-Cloning: the measurement byAlice leads to the collapse of the wave function and therefore to the lossof the initial state in her possession, respecting the No-Cloning theorem.
Quantum TeleportationQuantum Teleportation Protocol: Final CommentsExperimental realizations of the protocol: In 2020, a team of researchers used quantum teleportation over 44 kmof optical fiber - https://arxiv.org/abs/2007.11157 In 2017 the record for the implementation of the "ground-to-satellite"quantum teleportation protocol over a distance ranging from 500 kmup to 1,400 km - https://www.nature.com/articles/nature23675
Superdense Coding
Superdense CodingAlice and Bob pre-share a pair of entangled qubits.Alice wants to communicate two bits of information to Bobby sending a single qubit.Quantum communication channelABEQS
Superdense CodingAlice applies a certain local operation on the qubit in herpossession in order to encode two bits of informationEncoding00011110
Superdense CodingAlice sends her qubit to Bob through the quantum communicationchannel, hence qubit of information is communicated from Alice to BobQuantum communication channelAB
Superdense CodingAlice sends her qubit to Bob through the quantum communicationchannel, hence qubit of information is communicated from Alice to BobQuantum communication channelABSuperdense Coding occurs when Bob measures his two qubits in the"Bell“ basis to determine which state was prepared by Alice00:01:10:11:
Superdense CodingTeleportation vs Superdense CodingThe teleportation protocol can bethought of as an inverted version ofthe superdense coding protocol, in thesense that Alice and Bob "swap theirequipment".Superdense Coding Experiments: In 2017, a fidelity of 0.87 achieved with optical 03/PhysRevLett.118.050501 Nel 2018, High dimensional ququarts (states obtained in photon pairs via non-degeneratespontaneous parametric down-conversion) used to achieve a 0.98 /7/eaat9304
Quantum Cryptography
Quantum CryptographyPublic Key Cryptography: RSAPublic key:Known by all. Used by the sender toencrypt a secret messagePrivate key:Known to the owner only. Used bythe receiver to decrypt the message
Quantum CryptographyPublic Key Cryptography: RSA
Quantum CryptographyPublic Key Cryptography: RSAAn eavesdropperhas to factorize Nin order to breakthis cryptosystem.
Quantum CryptographyPublic Key Cryptography: RSAEasy example
Quantum CryptographyPublic Key Cryptography: RSAPublic key:Known by all. Used by the sender toencrypt a secret messagePrivate key:Known to the owner only. Used bythe receiver to decrypt the messageIn theory it is possible toextrapolate the private key
Quantum CryptographyPublic Key Cryptography: RSAIn order to obtain the privatekey, we need to solve a hardmathematical problemNumber ofoperationsLargest number everfactorized: 230 digitsFacorization of integer numbersN p x qRun-time best classical algorithm:𝒍𝒐𝒈(𝑵)𝟏/𝟑𝒆Length of N
Quantum Cryptography: Shor AlgorithmExponentialspeedupTime to factor a2048-digits number billions of years* seconds** Assuming we have a fault-tolerant quantum computer capable of executing Shor’s algorithm byapplying gates at the speed of current quantum computers based on superconducting circuits
Quantum CryptographyQuantum creates the problem but also provides the solutionQuantum MechanicsShor QuantumAlgorithmQuantumCryptography (QKD)Breaks RSAcryptograpySecure quantumcommunication
Quantum Key Distribution (QKD)
Quantum Key Distribution (QKD)Quantum key distribution is a system for ensuring secure communications. Itenables two parties to produce and share a random secret key only betweenthemselves which they can use to encrypt and decrypt their messages.The security of QKD relies on the fundamentals of quantummechanics compared to the traditional classical protocol whichis based on the computational hardness of certain mathematicalfunctions, and cannot provide any indications regardingpossible interceptions.
Quantum Key Distribution (QKD)Quantum key distribution is a system for ensuring secure communications. Itenables two parties to produce and share a random secret key only betweenthemselves which they can use to encrypt and decrypt their messages.An important and unique property of the QKD is the abilityof the two communicating users (Alice and Bob) to detect thepresence of a third party (Eve) who tries to obtaininformation on the secret key, due to the fact that ameasurement process disturbs the quantum system.
Quantum Key Distribution (QKD)Quantum key distribution is a system for ensuring secure communications. Itenables two parties to produce and share a random secret key only betweenthemselves which they can use to encrypt and decrypt their messages.Classical channelQuantum channel
Quantum Key Distribution (QKD)1. Alice uses two random bit (a, b) inorder to preare a qubit state 𝜓ab ۧ
Quantum Key Distribution (QKD)1. Alice uses two random bit (a, b) inorder to preare a qubit state 𝜓ab ۧ2. Alice sends the qubit 𝜓ab ۧ to Bobvia a quantum channel 𝜓ab ۧ
Quantum Key Distribution (QKD)3. Bob throws a random bit b′ to decide how to measurethe state 𝜓ab ۧ of the qubit that Alice transmitted
Quantum Key Distribution (QKD)3. Bob throws a random bit b′ to decide how to measurethe state 𝜓ab ۧ of the qubit that Alice transmitted4. Bob saves the result of a measurement in a bit a′a′ ቊ0 if outcome is 11 if outcome is 1
Quantum Key Distribution (QKD)After repeting steps 1,2,3 and 4 a number n of times,Alice e Bob publicly share their strings b e b′b′b
Quantum Key Distribution (QKD)After repeting steps 1,2,3 and 4 a number n of times,Alice e Bob publicly share their strings b e b′b′bThey discard all bits of thetwo strings except thosefor which b′ b
Quantum Key Distribution (QKD)After repeting steps 1,2,3 and 4 a number n of times,Alice e Bob publicly share their strings b e b′b′ba′ aThey discard all bits of thetwo strings except thosefor which b′ bThe remaining bits (asymptotically n / 2) will satisfy therelation a′ a and thus constitute their secret key.
Quantum Key Distribution (QKD)
Quantum Key Distribution (QKD)
Quantum Key Distribution (QKD)Let's imagine that Eve wants to intercept the secret key.Eve opts for an "Intercept-Resending" strategy in which she intercepts the qubit sentby Alice and measures it. She then sends back to Bob the state she measured.Classical channelQuantum channel
Quantum Key Distribution (QKD) If Eve uses the same basis used by Alice, then she can perfectlyunderstand the bit encoded by Alice, which will then be the same bitmeasured by Bob (if he also measures in the correct basis) If Eve uses a different basis from the one used by Alice, then her bit willbe random and so will the one measured by Bob.
Quantum Key Distribution (QKD)If Eve uses a different basis from the one used by Alice, then its bit will berandom as well as the bit measured by BobEve destroys the state if she doesn’t measure inthe correct basis
Quantum Key Distribution (QKD)If Eve uses a different basis from that used by Alice, then its bit will berandom as well as that measured by BobEve destroys the state if she doesn’t measure inthe correct basisa′aAlice and Bob extractpart of the secret keyand make it public
Quantum Key Distribution (QKD)a′aIf (asymptotically) ¼ ofthe bits of the publicsecret key are different,then they can claim thatEve is spying on them
Quantum Key Distribution (QKD)QKD Experiments:
Quantum CryptographyQuantum creates the problem but also provides the solutionQuantum MechanicsShor QuantumAlgorithmQuantumCryptography (QKD)Breaks RSAcryptograpySecure quantumcommunication
Quantum Key Distribution (QKD)Quantum creates the problem but also provides the solutionPost-Quantum CryptographySymmetric cryptographic algorithms and hashfunctions are considered safe from attacks byquantum computers. Lattice-based cryptography Multivariate cryptography Hash-based cryptographyNIST Call for dardization
Quantum Computing @ CINECACINECA: Italian HPC centerCINECA Quantum Computing Lab:- Research with Universities, Industries and QC startups- Internship programs, Courses and Conference mengoni@cineca.it
Postulates of Quantum Computing (1) To a closed quantum system is associated a space of states H which is a Hilbert space. The pure state of the system is then represented by a unit norm vector on such Hilbert space. The unit of quantum information is the quantum bit a.k.a. Qubit State of a qubit: Quantumly
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
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