Quantum Technology - Concepts and Prospectschep.iisc.ac.in/Personnel/adpatel/comput11.pdf · A....

Quantum TechnologyConcepts and Prospects

Apoorva D. Patel

Centre for High Energy Physics, Indian Institute of Science, Bangalorehttp://www.iisc.ac.in/initiative-on-quantum-technologies/

Meeting on Quantum Computation and Information26 September 2019, BARC, Mumbai

A. Patel (IISc) Quantum Technology 26 Sep 2019, BARC 1 / 22

Quantum Technologies

The field of quantum technologies is poised for significant breakthroughsin the coming years, implying life-changing consequences.The essential features that contribute to these technologies aresuperposition, entanglement, squeezing and tunneling of quantum states.




Practical applications are expected to appear• first in sensing and metrology,• then in communications and simulations,• then as feedback to foundations of quantum theory,• and ultimately in computation.





Developments in quantum technologies will also push classical technologiesin new directions. (“Quantum supremacy” is a moving target.)





Developments in quantum technologies will also push classical technologiesin new directions. (“Quantum supremacy” is a moving target.)

Many organizations have formulated detailed roadmaps.Europe roadmap: https://arXiv.org/abs/1712.03773USA roadmap: https://www.whitehouse.gov/sites/whitehouse.gov/files/images/

Quantum Info Sci Report 2016 07 22%20final.pdf

We need a similar roadmap for India.


Sensors and Metrology

High accuracy measurements have wide ranging applicationsfrom fundamental science to engineering and biology.Quantum precision is 1/N, compared to classical 1/

√N scaling of central limit theorem.


Sensors and Metrology

High accuracy measurements have wide ranging applicationsfrom fundamental science to engineering and biology.Quantum precision is 1/N, compared to classical 1/

√N scaling of central limit theorem.

Examples:

• Atomic clock precision can reach the Heisenberg limit.GPS accuracy can be improved by an order of magnitude.

• Nitrogen vacancy centres in diamond can be hyperpolarised,and function as highly sensitive and robust magnetometers.MRI scanners can be reducd to hand-held devices.

• Silicon vacancy centres in diamond have high coherence.They can be useful in forming quantum memories and networks.

• Multi-path atom interferometers can be used for precise inertialnavigation (measuring acceleration and rotation).This would bypass reliance on GPS networks. Adaption for use as gravimeters is possible.

• Precise electromechanical nanosensors can be made with 2D materials(involving both photons and phonons).Transducers have a variety of uses in sensing and control.


Quantum Communications

Focus areas:

• Superadditivity of channels to improve the signal to noise ratio.Wave nature makes {|ψ〉 ⊗ |ψ〉, |φ〉 ⊗ |φ〉} more distinguishable than {|ψ〉, |φ〉}.(Fourier transform of long exposures in astronomy produces high resolution images)

• Quantum random number generators.Use intrinisic randomness of quantum measurement. (Possible with mobile phone cameras)

• Post-quantum cryptography.Use quantum-hard problems to develop classical communication protocols that are secureagainst quantum attackers. (Lattice codes, Learning with errors)

• Heralded single photon sources.Frequency combs can accurately fix time window for individual photons.

• Efficient single photon detectors.Layered materials provide fast response and short dead time.

• Quantum imaging with entangled photons.Photon pair correlations increase the resolution beyond what is possible in classical optics.



Focus areas:







For the future: Quantum signal repeaters.Equivalent to quantum computers because error correction needs to be included.



Focus areas:







For the future: Quantum signal repeaters.Equivalent to quantum computers because error correction needs to be included.

Skip: Quantum security.========== (No practical advantage)


Quantum Computation

In going from classical to quantum computers, the concept of what iscomputable and what is not does not change, but the criteria ofcomputational efficiency do.Computation complexity is a measure of the physical resources requiredto solve a computational problem. It is specified in terms of the input sizeand the output precision.


Quantum Computation


Simulation problems are often expressed as time evolution under specifiedinteractions, from some simple initial state to the final state whoseproperties are to be determined.Quantum simulations can sum multiple evolutionary paths in superpositionat one go, while classical simulations need to evaluate them one by one.


Quantum Computation


Simulation problems are often expressed as time evolution under specifiedinteractions, from some simple initial state to the final state whoseproperties are to be determined.Quantum simulations can sum multiple evolutionary paths in superpositionat one go, while classical simulations need to evaluate them one by one.

In the 2n-dimensional Hilbert space of n qubits, we can superpose 2n

components evolving in parallel, but we can measure only n binaryobservables at the end. So the exponential gain of superposition is limitedby the restriction to extract only a small number of results at the end.

Efficient solutions exist only for problems with special structures.


Our Level of Understanding

The Advantage:

Efficient quantum algorithms have been discovered for specific problems,which are believed to be hard for classical computers.Easily prepared entangled quantum states have superclassical correlations.Their correlated probability distributions cannot be efficiently sampled byany classical (stochastic) means.There is no efficient classical algorithm to simulate a quantum computer.



The Advantage:


The Target:

Study problems that are BQP (Bounded error Quantum Polynomial) hard.There exist problems that are even harder for quantum computers (i.e. QMA-hard).



The Advantage:


The Target:

Study problems that are BQP (Bounded error Quantum Polynomial) hard.There exist problems that are even harder for quantum computers (i.e. QMA-hard).

Nature produces highly entangled many-body quantum states, and it ishard to study their properties classically, due to the “sign problem”. (e.g.real time interactions of elementary particles, quantum behaviour of blackholes, evolution of early universe, strongly correlated electron systems)Their complexity is high due to non-equilibrium, quantum chaos or information scrambling.If we have understood the laws of nature, then we should be able to duplicate them.


The Hurdle

Quantum systems are highly sensitive to disturbances from theenvironment. Even necessary observations perturb them.

The disturbances can be viewed as unwanted scatterings. They can besuppressed by reducing the flux (e.g. low temperatures), reducing thecoupling (e.g. neutral degrees of freedom), or reducing the final phasespace (e.g. gap in spectrum).But the noise brought in by the controls needed to manipulate the systemcannot be avoided, and has to be handled differently.


The Hurdle

Quantum systems are highly sensitive to disturbances from theenvironment. Even necessary observations perturb them.

The disturbances can be viewed as unwanted scatterings. They can besuppressed by reducing the flux (e.g. low temperatures), reducing thecoupling (e.g. neutral degrees of freedom), or reducing the final phasespace (e.g. gap in spectrum).But the noise brought in by the controls needed to manipulate the systemcannot be avoided, and has to be handled differently.

Encoded quantum information can be protected by separating the scales oferror and the code (e.g. error correction or decoherence-free subspaces).That can lead to scalable and fault-tolerant computation, but only whenthe error rate is below a certain threshold. The present hardwaretechnology is orders of magnitudes away from reaching such thresholds.

Techniques to bring down the environmental error rate, and to find systemrealisations with high fault-tolerance thresholds, are crucial and are beingintensively pursued.


Hardware Design Criteria

Experimental setups need to meet the following conditions, in order toperform as reliable quantum computers. —D.P. DiVincenzo




• A scalable physical system with well characterized qubits.

• The ability to initialize the state of the qubits to a simple fiducial state(e.g. the ground state).

• Long decoherence time compared to logic operation time.

• An addressable universal set of quantum gates.

• A qubit-specific measurement capability.

• The ability to interconvert stationary and flying qubits(for communication between CPU and memory).

• The ability to faithfully transmit flying qubits between specified locations(as replacement for wires).




• A scalable physical system with well characterized qubits.

• The ability to initialize the state of the qubits to a simple fiducial state(e.g. the ground state).

• Long decoherence time compared to logic operation time.

• An addressable universal set of quantum gates.

• A qubit-specific measurement capability.

• The ability to interconvert stationary and flying qubits(for communication between CPU and memory).

• The ability to faithfully transmit flying qubits between specified locations(as replacement for wires).

Error correction needs discrete variables

with error rate below a particular threshold.


Explorations

Discrete variables:Qubit TechnologyElectron spin Crystal defects, DopingNuclear spin Nuclear magnetic resonancePhoton path and polarisation Quantum optics, Cavity QEDTwo-level atom Ion traps, Quantum dotsUnusual/artificial atoms Rydberg atoms, TransmonsMagnetic flux quantum Josephson junction circuitsNon-abelian anyon Topological materials

Continuous variables:Bose-Einstein condensates, Adiabatic quantum evolution.


Explorations

Discrete variables:Qubit TechnologyElectron spin Crystal defects, DopingNuclear spin Nuclear magnetic resonancePhoton path and polarisation Quantum optics, Cavity QEDTwo-level atom Ion traps, Quantum dotsUnusual/artificial atoms Rydberg atoms, TransmonsMagnetic flux quantum Josephson junction circuitsNon-abelian anyon Topological materials

Continuous variables:Bose-Einstein condensates, Adiabatic quantum evolution.

The number of physical qubits in a quantum device is approximatelydoubling every year (exceeds Moore’s law).Dedicated research centres have been funded at the level of 50-100 million US dollars.

Current status: https://quantumcomputingreport.com/scorecards/


https://quantumcomputingreport.com/scorecards/

Quantum Information Processing

•••

The challenge is much more than just scaling up.M.H. Devoret and R.J. Schoelkopf, Science 339 (2013) 1169-1174


Quantum Information Processing

•••

The challenge is much more than just scaling up.M.H. Devoret and R.J. Schoelkopf, Science 339 (2013) 1169-1174

Quantum simulations would model physical systems directly into quantum hardware, but withgreater freedom in the choice of parameters than the limited values the natural systems have.This is the likely area where quantum supremacy will be demonstrated.


Quantum Algorithms

• Periodic patterns are easily found by Fourier Transform, which is afamiliar unitary operation in quantum theory.Shor’s factorisation algorithm uses quantum Fourier transform for exponential speed up.


Quantum Algorithms


• Grover’s quantum search algorithm for an unsorted database executes adirected search along a geodesic.A single target among N items can be located with O(

√N) oracle queries.


Quantum Algorithms




• Random walks represent a diffusion process.Classical diffusion operator is the Laplacian: ∂P

∂t = ∇2P .E(~k) ∝ |~k|2 produces the characteristic Brownian motion signature: distance ∝

√time


Quantum Algorithms






√time

Relativistic quantum evolution with E(~k) ∝ |~k| produces the signature: distance ∝ time

Any NP-complete problem speeds up at least quadratically.


Quantum Algorithms






√time

Relativistic quantum evolution with E(~k) ∝ |~k| produces the signature: distance ∝ time

Any NP-complete problem speeds up at least quadratically.

• Multi-fermion wavefunctions are totally antisymmetric determinants(easy to compute). Multi-boson wavefunctions are completely symmetricpermanents (hard to compute).Boson sampling with n identical photons naturally generates n!-component symmetric state.


Grover Search (An illustration)

The key feature of the algorithm is wave dynamics, and not entanglement.Using a single oracle call, the algorithm identifies 1 out of 4 items in thedatabase. In contrast, a Boolean algorithm identifies only 1 out of 2 items.

Amplitudes Algorithmic Steps Physical Implementation

(1) 0

0.5Uniformdistribution

Equilibriumconfiguration

(The first item is desired by the oracle. The dashed line denotes the average amplitude.)





(1) 0



❄Ub Quantum oracle Binary question

(2) 00.25 Amplitude of

desired stateflipped in sign

Suddenperturbation






(1) 0






Suddenperturbation

❄

−Us Reflectionabout average

Overrelaxation






(1) 0






Suddenperturbation

❄

−Us Reflectionabout average

Overrelaxation

(3)

00.25

♣ ♣ ♣

Desired statereached

Opposite end

of oscillation

(4) Projection Algorithmis stopped

Measurement



Possible uses

Grover’s algorithm is an amplitude amplification process.A system of coupled wave modes can execute it, provided(1) Superposition of modes maintains phase coherence.(2) The two reflection operations (phase changes of π for theappropriate mode) can be efficiently implemented.


Possible uses


In the quantum version, |A|2 gives the probability of a state, and thealgorithm solves the database search problem.In the classical wave version, |A|2 gives the energy of a mode, and thealgorithm provides the fastest method for energy redistribution throughthe phenomenon of beats.


Possible uses


In the quantum version, |A|2 gives the probability of a state, and thealgorithm solves the database search problem.In the classical wave version, |A|2 gives the energy of a mode, and thealgorithm provides the fastest method for energy redistribution throughthe phenomenon of beats.

• Focusing of energy can be used as a selective switch.• Energy amplification can speed up catalytic processes.• Fast dispersal of energy can be used in shock absorbers.• Defects and impurities in materials can be detected bywave reflections at suitably tuned frequencies.


Genetic languages

1 What is the information processing task carried out by the geneticmachinery of every living organism?Assembling molecules by picking up components from an unsorteddatabase.

2 What is the optimal way of carrying out this task?Lov Grover’s quantum search algorithm. (Requires wave dynamics.)

3 What is the signature of this algorithm?

(2Q + 1) sin−1 1√N

=π

2=⇒

Q = 1, N = 4

Q = 2, N = 10.5

Q = 3, N = 20.2


Genetic languages




(2Q + 1) sin−1 1√N

=π

2=⇒

Q = 1, N = 4

Q = 2, N = 10.5

Q = 3, N = 20.2

Molecular biology is a nanotechnology that has worked for billions of yearsand in an amazing variety of circumstances. Darwinian evolution hasoptimised its basic processes to their essentially universal forms.

Classically, two nucleotide bases (one complementary pair) are sufficient to encodethe genetic information. What led nature to complicate this simpler system?


Genetic languages




(2Q + 1) sin−1 1√N

=π

2=⇒

Q = 1, N = 4

Q = 2, N = 10.5

Q = 3, N = 20.2

Molecular biology is a nanotechnology that has worked for billions of yearsand in an amazing variety of circumstances. Darwinian evolution hasoptimised its basic processes to their essentially universal forms.

Classically, two nucleotide bases (one complementary pair) are sufficient to encodethe genetic information. What led nature to complicate this simpler system?

Telltale signatures of wave dynamics (vibronic modes) are present in:Enzyme catalysis, photosynthesis, olfaction, magnetoreception by birds.


The NISQ Era

Digital Systems:

Noisy Intermediate Scale Quantum systems (10-100 qubits) are becomingavailable. They are special purpose platforms, likely to be accessible via thecloud. Current fastest classical computers can simulate around 50 qubits.The fault-tolerance threshold is orders of magnitudes of away, so there is no error correction.


The NISQ Era

Digital Systems:


The interactions between qubits is limited (nearest neighbour), and about1000 logic gate operations may be performed within the coherence time.A validation and verification method may not exist for simulation problems.


The NISQ Era

Digital Systems:



Analog systems:

These resemble the physical systems more closely, and may work wellempirically, especially while investigating universal properties (e.g. trappedions for spin systems, cold atom condensates for chemical reactions).Control of annealing and adiabatic evolution is error-prone (signal and error are mixed together).


The NISQ Era

Digital Systems:



Analog systems:

These resemble the physical systems more closely, and may work wellempirically, especially while investigating universal properties (e.g. trappedions for spin systems, cold atom condensates for chemical reactions).Control of annealing and adiabatic evolution is error-prone (signal and error are mixed together).

Hope:

Even in absence of a rigorous proof, some heuristic algorithms may getdiscovered, as in case of the simplex method and the neural networks.There is need for trial-and-error explorations, both in software and hardware. NISQ studieswill not transform the world, but will offer tremendous opportunities for the next stage.


Directions to Explore

Hybrid quantum-classical systems:

These are well-suited to solve optimisation problems, with a quantumprocessor and a classical optimiser. The classical feedback alters the inputquantum parameters until the result converges.e.g. Quantum Approximate Optimisation Algorithms for combinatorial optimisation problems,Variational Quantum Eigensolvers for low-lying states of many-body quantum systems.Small molecules and nuclei have been studied using the Unitary Coupled Cluster method.





Noise-resilient circuits:

The error bound 1/G , for a logic ciruit containing G gates, is pessimistic.Errors often affect only nearby algorithmic steps, and also decay away.Universality and variational principles stabilise physical simulations.Careful design and characterisation of quantum logic circuits can help tremendously.





Noise-resilient circuits:

The error bound 1/G , for a logic ciruit containing G gates, is pessimistic.Errors often affect only nearby algorithmic steps, and also decay away.Universality and variational principles stabilise physical simulations.Careful design and characterisation of quantum logic circuits can help tremendously.

Linear algebra problems:

Ax = b can be efficiently solved by the Newton-Raphson method.eA can be efficiently evaluated using the Chebyshev polynomial expansion.Sparse matrices with block-diagonal decomposition are a must. Preconditioning is desirable.


Quantum deep learning:

Layered systems of “neurons” are good at parsing large amount of datausing simple constraints. Labeled data sets can be used to train thenetwork to identify specific patterns. A quantum version is needed toextract desired features from Quantum Random Access Memory.2n entangled components of an n-qubit state has only n bits of measurable information.Can quantum neurons be trained to efficiently recognise complex entanglement patterns?




Recommendation systems:

Big data analysis often needs value judgements, given limited knowledgeof preferences, while looking for a small number of pattern types (i.e. thepreference matrix has a low rank). Online recommendations are made,after constructing an offline approximation to the preference matrix.Quantum algorithms are better at sampling the low-rank approximate preference matrix.




Recommendation systems:

Big data analysis often needs value judgements, given limited knowledgeof preferences, while looking for a small number of pattern types (i.e. thepreference matrix has a low rank). Online recommendations are made,after constructing an offline approximation to the preference matrix.Quantum algorithms are better at sampling the low-rank approximate preference matrix.

Semidefinite programming:

The task is to optimise a linear function subject to some matrix inequalityconstraints, i.e. maximise Tr(CX ) while satisfying Tr(AiX ) ≤ bi forHermitian matrices. This convex optimisation has many applications.In a quantum setting, the problem maps to preparation of a density matrix in a Gibbs state.Efficient preparation of the Gibbs state is possible when the system thermalises rapidly.


Software Quantum Simulator

Functioning of imperfect devices requires validation and verification.Programs running on classical parallel computer clusters can simulate10-50 qubit systems, which is the size of NISQ quantum processors.




Instead of using exact Boolean algebra, the simulator can be designed tomimic a noisy quantum processor. The evolution has to be described inthe density matrix formalism, incorporating various sources of error:

• Thermal effects in initialistion of the quantum register• Imperfect implementation of the logic gates• Decay and decoherence of quantum states in memory• Depolarisation during measurement

The simulator can also be restricted in connectivity between qubits,to imitate what may be the structure of a real quantum processor.




Instead of using exact Boolean algebra, the simulator can be designed tomimic a noisy quantum processor. The evolution has to be described inthe density matrix formalism, incorporating various sources of error:

• Thermal effects in initialistion of the quantum register• Imperfect implementation of the logic gates• Decay and decoherence of quantum states in memory• Depolarisation during measurement

The simulator can also be restricted in connectivity between qubits,to imitate what may be the structure of a real quantum processor.

Such a simulator can test how well various algorithms work on imperfectquantum processors. More importantly, one can vary the imperfectionsand connectivity in the software, to figure out what design for the noisyquantum processor would produce the best results.


References

M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information,Cambridge University Press (2000)

J. Preskill, Quantum Computing in the NISQ Era and Beyond, Quantum 2 (2018) 79J. Preskill, Simulating Quantum Field Theory with a Quantum Computer,

LATTICE2018 Proceedings [arXiv:1811.10085]A. Patel and A. Priyadarsini, Efficient Quantum Algorithms for State Measurement and Linear

Algebra Applications, Int. J. Quantum Inform. 16 (2018) 1850048R.P. Brent and P. Zimmermann, Modern Computer Arithmetic,

Cambridge University Press (2010)H. Choudhary, B. Mahato, L. Priyadarshi, N. Roshan, Utkarsh and A.D. Patel,

A Software Simulator for Noisy Quantum Circuits, arXiv:1908.05154


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