Obfuscation Quantum Obfuscation Triorthogonal matrix Magic state distillation

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Obfuscation Quantum Obfuscation Triorthogonal matrix Magic state distillation

Harris, Mark, Contributing Editor has reference to this Academic Journal, PHwiki organized this Journal Research on Quantum Algorithms at the Institute as long as Quantum In as long as mation in addition to MatterJ. Preskill, L. Schulman, Caltechpreskill@caltech.edu / www.iqim.caltech.edu/ObjectiveObjective Approach Improved rigorous estimates of thresholds as long as fault-tolerant quantum computation. Quantum algorithms beyond the hidden subgroup paradigm. Quantum in addition to classical simulation methods as long as quantum many-body systems. New approaches to physically robust quantum computation. Quantum algorithms as long as simulating local quantum systems. Novel applications of the quantum Fourier trans as long as m in addition to other trans as long as ms. Customizing quantum fault tolerance as long as physically motivated noise models. Schemes as long as physically robust quantum storage in addition to processing. Characterizing Hamiltonian complexity. Quantum-resistant classical cryptography.StatusQuantum circuit obfuscation schemes based on the connections between quantum circuits in addition to braids.Proposed quantum-resistant cryptosystem based on hardness of solving systems of quadratic equations.Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes.Scheme as long as per as long as ming protected quantum gates based on a continuous-variable quantum codes.Sufficient condition on noise correlations as long as scalable quantum computing. Magic state distillation with low overhead. Research plan as long as the next 12 months – FY13-14Quantum algorithms as long as simulating quantum field theories with gauge fields in addition to massless particles. Quantum algorithms as long as simulating thermalization of quantum systems. Quantum algorithms as long as interpolating b in addition to -limited functions on continuous groups.Renormalization group analysis of three-dimensional topological quantum codes.Probability distributions that can be sampled efficiently quantumly but not classically.Structurally inhomogeneous tensor network states as long as strongly disordered systems. Long term objectives- Bring large-scale quantum computers closer to realization by proposing in addition to analyzing new schemes as long as protecting quantum systems from noise.- Conceive, develop, in addition to analyze new applications of quantum computing to physics in addition to mathematics. Progress on last year’s objectives – FY12-13Quantum algorithms as long as simulating particle collisions in fermionic quantum field theories.Proposed quantum-resistant cryptosystem based on hardness of solving systems of quadratic equations.Quantum circuit obfuscation schemes based on the connections between quantum circuits in addition to braids.Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes.Efficient algorithm as long as testing stability of two-dimensional tensor-network states vs. local perturbations. Scheme as long as per as long as ming protected quantum gates based on a continuous-variable quantum codes.Sufficient condition on noise correlations as long as scalable quantum computing. Near-optimal dynamical decoupling schemes as long as multi-level quantum systems.New class of highly entangled many-body states which can be efficiently simulated.Research on Quantum Algorithms at the Institute as long as Quantum In as long as mation in addition to MatterJ. Preskill, L. Schulman, Caltechpreskill@caltech.edu / www.iqim.caltech.edu/Students:Michael Beverl in addition to Peter Brooks HRLBill FeffermanJeongwan Haah MITIsaac Kim PerimeterAlex Kubica Shaun Maguire Evgeny MozgunovSujeet ShuklaUndergrads (4 in 2012, 3 in 2013)Visitors:ManyPostdocs arriving 2013-14:Mario Berta (ETH)Andrew Essin (Colorado)O. L in addition to on-Cardinal (Sherbrooke)Kristan Temme (MIT) Faculty: John Preskill Alexei Kitaev Leonard Schulman Gil Refael Faculty Associates:Todd Brun Steven van Enk S in addition to y Irani Postdocs:Gorjan AlagicGlen EvenblyAlexey Gorshkov NISTZhengcheng Gu PerimeterNate Lindner TechnionSpiros MichalakisFern in addition to o PastawskiLing WangBeni YoshidaDaniel LidarKirill ShtengelResearch on Quantum Algorithms at the IQIM

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Some research themes at the IQIM Power of quantum computing. Simulating quantum field theories, preparing thermal states, obfuscating quantum circuits with braids, quantum-resistant public key based on multivariate quadratic equations, quantum algorithms as long as interpolating b in addition to -limited functions on continuous groups, probability distributions that can be sampled efficiently quantumly but not classically.Fault-tolerant quantum computing. Magic-state distillation protocol using triorthogonal quantum codes, RG analysis of self-correcting quantum memory in 3D, universal topological quantum computing with realistic materials, protected gates as long as superconducting qubits, universal dynamical decoupling, asymmetric Bacon-Shor codes as long as protection against biased noise. Experiment in addition to implementation. Attractive photons in a quantum nonlinear mediua, Kitaev honeycomb in addition to other exotic spin models with polar molecules, realizing fractional Chern insulators with dipolar spins. Quantum many-body physics. Classifying locally definable quantum phases, area law in addition to sub-exponential algorithm as long as 1D systems, fractional Majorana fermions at the edges of abelian quantum Hall states, class of highly entangled many-body states which can be efficiently simulated, structurally inhomogeneous tensor network states as long as strongly disordered systems.Quantum algorithms as long as quantum field theories- Feynman diagrams have limited precision, particularly at strong coupling.- Classical lattice methods can compute static properties, but cannot simulate dynamicsA quantum computer can simulate particle collisions, even at high energy in addition to strong coupling, using resources (number of qubits in addition to gates) scaling polynomially with precision, energy, in addition to number of particles. – Estimate errors due to regulating (spatial lattice in addition to approximating continuous variable fields by qubits).- Efficient procedure as long as preparing (strongly-coupled) vacuum in addition to initial wave packet states, simulating time evolution, measuring final state. Does the quantum circuit model capture the computational power of NatureWhat about quantum gravityJordan, Lee, PreskillSimulating quantum field theoryInput: a list of incoming particle momenta (particles are actually wave packets with some momentum spread).Output: a list of outgoing particle momenta.Goal is to sample accurately from the distribution of final state particles that would be produced in a high energy collision in a (strongly coupled) field theory.Previous work: Consider a self-coupled scalar field in d = 1, 2, 3, spatial dimensions. Digitize field at each lattice point using nb qubits, where nb scales logarithmically with energy in addition to accuracy.Procedure:Prepare free field vacuum.Prepare free field wavepackets.Adiabatically turn on the coupling constant (t).Evolve as long as time T using interacting Hamiltonian H.Adiabatically turn off couplingMeasure field modes of free theory. Need to discretize the problem, in addition to keep track of resulting errors. Jordan, Lee, Preskill

Simulating fermionic quantum field theoryInput: a list of incoming particle momenta (particles are actually wave packets with some momentum spread).Output: a list of outgoing particle momenta.Goal is to sample accurately from the distribution of final state particles that would be produced in a high energy collision in a (strongly coupled) field theory.This year’s work: Consider a self-coupled fermionic field in d = 1 spatial dimensions (e.g., Gross-Neveu model). Procedure:Prepare uncoupled fermion modes.Adiabatically turn on nearest neighbor coupling between modes.Adiabatically turn on the coupling constant (t).Excite spatially localized wave packets with time-dependent sources.Measure charge in addition to postselect on detecting one particle.Evolve as long as time T using interacting Hamiltonian H.Nondestructively measure energy in addition to momentum of outgoing particles.Need to discretize the problem, in addition to keep track of resulting errors. Jordan, Lee, PreskillSimulating fermionic quantum field theoryFree fermion vacuum is not Gaussian – prepare it by adiabatically turning on nearest neighbor coupling between modes.Fermi minus sign: Use Bravyi-Kitaev encoding at cost O(log L). When a fermionic gate is applied, relative sign of 0> in addition to 1> depends on occupation numbers of other modes (e.g. the number of occupied modes to the left of the given site). We could represent fermion operators as (Jordan-Wigner) nonlocal string operators at cost O(L), or we could store the partial sums of mode occupation numbers, but then updates have cost O(L). Better: cleverly choose partial sums which allow computation of (-1)’s in O(log L) in addition to can be updated in time O(log L). Exciting wave packets: Modulate source spatially in addition to temporally to match one particle states. Make the source weak to avoid creating more than one particle, but it usually produces nothing. Measure in addition to abort if not particle created (okay as long as a collision of a constant number of particles).Advantage over previous method (in which coupling ramps on after wavepacket created): works as long as bound states. Jordan, Lee, PreskillSimulating fermionic quantum field theoryProcedure:Prepare uncoupled fermion modes.Adiabatically turn on nearest neighbor coupling between modes.Adiabatically turn on the coupling constant (t).Excite spatially localized wave packets with time-dependent sources.Measure charge in addition to postselect on detecting one particle.Evolve as long as time T using interacting Hamiltonian H.Nondestructively measure energy in addition to momentum of outgoing particles.Need to discretize the problem, in addition to keep track of resulting errors. Cost is dominated by the adiabatic preparation of the vacuum. Adiabaticity en as long as ces turn-on timeJordan, Lee, Preskillwhere a is the lattice spacing in addition to is the error. Using a high-order Trotter approximation, the number of gates needed is: The error due to nonzero lattice spacing scales as ~ a, hence cost scales with error as (seems pessimistic)

Simulating quantum field theoryFuture plans:Massless particles (infrared safe observables).Gauge fields (start with strong coupling limit).Ground state preparation by cooling.Nonzero temperature in addition to chemical potential.Simulate st in addition to ard model of particle physics in BQP.Quantum gravityObfuscationTake a circuit C in addition to produce another circuit O(C), so that:functionality is preserved;size is not much bigger (say polynomial);it’s hard to “reverse-engineer” O(C) (at a minimum, O(C) -> C is hard).Can we have an algorithm that does this as long as all circuitsState of affairs in researchlots of motivation (software/hardware copy protection, homomorphic encryption, turning private key schemes into public key schemes, etc.)known as long as malizations of (3) are all too hard:O(C) no more useful than a black box that per as long as ms C (impossible, Barak et al ’01)O(C1) indistinguishable from O(C2) as long as equivalent C1, C2 (collapses PH, Goldwasser Rothblum ’07)little is known about quantum obfuscationare there classical algorithms as long as obfuscating quantum circuitsare there quantum states that allow us to do obfuscated computationG. Alagic, T. Jeffery,S. JordanQuantum ObfuscationWhat if we ask as long as a slightly weaker condition (3) Can we obfuscate quantum circuitsResults [Alagic Jeffery Jordan ’13]efficient classical algorithms as long as obfuscating both quantum in addition to classical circuits“weaker” condition 3: indistinguishability under a subset of the set of all circuit relationsCore ideaif we had an efficient canonical as long as m as long as circuits (a coNP-hard problem), we would satisfy Goldwasser-Rothblum triviallybut topological quantum computation gives us a pretty good mapping in addition to braids do have efficient canonical as long as ms!in fact, this mapping exists as long as classical reversible circuits too, if we use a different representation of the braid groupIf Bob claims to have a quantum computer, Alice can propose that Bob execute a quantum circuit that obfuscates a classical circuit, where Alice can easily check the answer. G. Alagic, T. Jeffery,S. Jordan

Approximation theory on groupsThe Discrete Fourier Trans as long as m (DFT) basis of countless proofs, algorithms, signal processing tasks, etc.the fast classical (FFT) algorithms as long as computing the DFT are very useful in practicetheir quantum analogues (QFT) are exponentially faster (in a certain sense) in addition to are a basis as long as amazing things like Shor’s algorithmWhat if the group is continuous instead of finite (say the circle or SU(2))finitely many sums becomes infinitely many integrals.two simplifications: only consider b in addition to -limited f (doesn’t oscillate too much), in addition to sample the function at a nicely spaced finite set of points as long as the circle, this boils down to “discretize in addition to use DFT”G. Alagic, A. Russell,L. SchulmanApproximation theory on groupsNew feature of continuous case: We can use Fourier inversion to reconstruct the values of the function anywhere on the group.Why study the continuous non-abelian casesignals in practice might be continuous instead of discretewe care about nonabelian spaces (e.g., spherical harmonics, SU(2))we need more quantum-algorithmic primitives as long as exponential speedupsResults [Alagic Russell Schulman 2013]a theorem about reconstructing b in addition to -limited functions on compact groupssetting: any compact groupinput: r in addition to om samples of a b in addition to -limited function foutput: the list of Fourier coefficients of fa number of samples cubic in the b in addition to limit is sufficient as long as a good estimatethe reconstruction is inner-product-preserving in the limit(Multivariate Quadratic + Code)-Based Cryptosystem Post-Quantum: honest players are classical in addition to polynomial-time, but adversary might have a quantum computer.Adversary knows public key, needs to solve a hard problem to decrypt (invert a one-way function). Private key provides a trap door as long as efficient decryption.Preferably based on a problem which is (average case) hard. Problem has structure which enables the trap door, but is hidden from the adversary.Preferably a simple scheme, so potential attacks are obvious — no well hidden vulnerabilities.Schulman

Post-quantum cryptographyNumber theoretic (abelian hidden subgroup problems): vulnerable to quantum attacks. RSA, elliptic curve, Diffie-Hellman, etc.Lattice cryptosystems: based on hardness of shortest/closest vector problems. Worst-case to average case reduction. Reduce to dihedral (nonabelian) hidden subgroup. Reasons as long as concern: – Single-register coset measurements info. theoretically sufficient. – Kuperberg algorithm: Time McEliece: based on hardness of decoding general linear EC codes.Public key C = M G PG generates linear code, M is r in addition to om matrix, P is r in addition to om permutation.Encode v as vC + correctable errors (weight t).Code is efficiently decodable, in addition to has to be carefully chosen. New proposal: a code-based scheme in which the scrambled code is not public.Schulman(Multivariate Quadratic + Code)-Based CryptosystemPublic: three-index 2N 2N L binary tensorR is r in addition to om 2N 2N K binary tensor , {a, b} are 2L r in addition to om length-(2N) vectors, in addition to C is generator of a scrambled length-L efficiently decodable linear code, which can correct most errors of weight as long as some > ¼ L.Clear text: Length-N binary string x.Encrypted text: Length-L binary string y (L > 8N). Append length-N r, s to x.where yl(error))=1 with probability ¼ (product of two r in addition to om bits).Decryption: Decode to find yl(error)). Each l as long as which yl(error))=1 provides linear equations as long as xr in addition to xs. 2N such equations suffice (hence L > 8N). Security: Here the scrambled code is hidden by the noise. (Known attacks on McEliece use the publicly known scrambled code.) Adversary needs to solve a r in addition to om system of quadratic equations to find x, if unable to infer structure of the public tensor T. To ensure hardness of tensor decomposition, dual of C should have positive fractional distance. Schulman(Multivariate Quadratic + Code)-Based Cryptosystem- Codes with the desired properties are not known.- One way around this is to use higher-order tensors; e.g. with an 4-index tensor we can reduce the correctable error rate of C to 1/8 (still requiring the dual to have positive fractional distance), in addition to then minimum distance decoding is feasible (but still no known codes). With an 7-index tensor the correctable error rate becomes 1/64, in addition to suitable efficiently decodable codes have been constructed by Guruswami 2009.- That means a larger public key, but the key size can be reduced somewhat by linearly hashing down the extra dimensions until their size is proportional to the security parameter.- Basing security on the hardness of tensor decomposition is a new feature in public key cryptography.Schulman

Protected superconducting qubit Two states localized near =0 in addition to = are the basis states of a protected qubit. The barrier is high enough to suppress bit flips, in addition to the stable degeneracy suppresses phase errors. Protection arises because the encoding of quantum in as long as mation is highly nonlocal, in addition to splitting of degeneracy scales exponentially with size of the device.Feigel’man & IoffeDoucot & VidalKitaev“0-Pi qubit”:Physically robust encodings have been proposed using superconducting circuits containing Josephson junctions, as long as example the “0-Pi qubit”. The circuit’s energy E(), as a function of the superconducting phase difference between its leads, is a periodic function with period to an excellent approximation.0Brooks, Kitaev, PreskillFor reliable quantum computing, we need not just very stable qubits, but also the ability to apply very accurate nontrivial quantum gates to the qubits. Accurate (Clif as long as d group) phase gates can be applied to 0-Pi qubits by turning on in addition to off the coupling between a qubit (or pair of qubits) in addition to a harmonic oscillator (an LC circuit whose inductance is large in natural units). In principle the gate error becomes exponentially small as the inductance grows.The reliability of the gate arises from a continuous-variable quantum error-correcting code underlying its operation, in which a qubit is embedded in the infinite-dimensional Hilbert space of a harmonic oscillator. Coupling the 0-Pi qubit to the oscillator sends the oscillator on a state-dependent phase space excursion during which it acquires a geometric phase that is protected by the code. Protected phase gate Brooks,Kitaev,PreskillSwitch is really a tunable Josephson junction:Under suitable adiabaticity conditions, closing the switch trans as long as ms a broad oscillator state (e.g. the ground state) into a grid state (approximate codeword).D-1V()Peaks are at even or odd multiples of depending on whether is 0 or , i.e. on whether qubit is 0 or 1. Inner width squared is (JC)-1/2 in addition to outer width is (L/C)1/2Protected phase gate

calculable contribution to error due to diabatic effects in addition to Q-space spreadingLarge inductance Manucharyan et al. 2009, Masluk et al. 2012, Bell et al. 2012 achieved ~ 20 with a chains of Josephson junctions. The inductance scales linearly with the length of the chain, but there are potential obstacles to building very long chains. Another possible approach is to exploit the large (kinetic) inductance in amorphous superconductors.The intrinsic error scales like What about universal quantum computation in addition to measurement- If we can prepare in addition to measure in the basis 0 ± 1, a noisy /4 single-qubit phase gate (F > .93), augmented by state distillation, suffices as long as fault-tolerant universality (Bravyi & Kitaev 2005).- It is also okay if measurements are noisier than gates, as we can protect measurements using repetition (or coding)- So if we can really do a two-qubit phase gate with high fidelity, that’s worth a lot!Magic State Distillation with Improved Overhead – In typical protocols as long as fault-tolerant quantum computing based on stabilizer codes, Clif as long as d operations (e.g. CNOT gates in addition to 90 degree single-qubit rotations) have relatively low overhead cost. – Overhead tends to be dominated by non-Clif as long as d operations, such as 45 degree single-qubit rotations, Toffoli (controlled-controlled-NOT) gates, or controlled-controlled phase gates. – For “magic-state distillation” protocols, we use codes such that the 45 degree rotation T is transversal. Triorthogonal codes admit such transversal logical gates.- For logical non-Clif as long as d gates with error rate , the cost scales like log(1/), in addition to we would like to reduce the exponent .- Exponent is = log(r) / log(a) if protocol yields one output copy as long as each r input copies, in addition to reduces error from p to O(pa).- New family of protocols asymptotically achieves = log(3) / log(2) ~ 1.6. Best previous protocol had been = log(5) / log(2) ~ 2.3.Bravyi, Haah

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Triorthogonal matrixA binary matrix where any pair of rows has even overlap in addition to so does any triple.E.g.Even-weight rows shown in bold.Number of odd-weight rows determines number k of encoded qubits in corresponding CSS code.Family of codes with length n = 3k + 8. Codes have distance d = 2.Magic state distillationBy a stabilizer code based on “triorthogonal matrices”Noisy magic states are represented by a stochastic application of Z-rotationDistills Pi/8-rotation magic stateDistillation cost improvedUsing a new explicit family of triorthogonal matrices G(k),Error rate improves asAvg. of input states to reach a target error rate isNumerical optimization, combining various protocols.At Target error rate 10-12:- 2-fold improvement from Meier-Eastin-Knill (1204.4221)- 10-fold improvement from (original) Bravyi-Kitaev (2004)In plot, upper curve is Meier et al.Lower curve is new protocol.

Entanglement RenormalizationLocal unitary trans as long as mation,On nearest neighborsfactoring out trivial degrees of freedom.Underst in addition to “long-range entanglement”HaahLaurent polynomial representation of stabilizer code HamiltoniansLocal unitary = row operationTrivial qubit = presence of sole 1 in a columnCoarse-graining = matrix expansionToric code maps to self in a coarse-graining stepCubic codeCubic code factorizes to itself plus another.The other factorizes into two copies of itself. (X-type stabilizers are shown.)2323Coarse-graining step:A A + BB B + B

Research plan as long as the next 12 months – FY13-14Quantum algorithms as long as simulating quantum field theories with gauge fields in addition to massless particles. Quantum algorithms as long as simulating thermalization of quantum systems. Quantum algorithms as long as interpolating b in addition to -limited functions on continuous groups.Renormalization group analysis of three-dimensional topological quantum codes.Probability distributions that can be sampled efficiently quantumly but not classically.Structurally inhomogeneous tensor network states as long as strongly disordered systems. Long term objectives- Bring large-scale quantum computers closer to realization by proposing in addition to analyzing new schemes as long as protecting quantum systems from noise.- Conceive, develop, in addition to analyze new applications of quantum computing to physics in addition to mathematics. Progress on last year’s objectives – FY12-13Quantum algorithms as long as simulating particle collisions in fermionic quantum field theories.Proposed quantum-resistant cryptosystem based on hardness of solving systems of quadratic equations.Quantum circuit obfuscation schemes based on the connections between quantum circuits in addition to braids.Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes.Efficient algorithm as long as testing stability of two-dimensional tensor-network states vs. local perturbations. Scheme as long as per as long as ming protected quantum gates based on a continuous-variable quantum codes.Sufficient condition on noise correlations as long as scalable quantum computing. Near-optimal dynamical decoupling schemes as long as multi-level quantum systems.New class of highly entangled many-body states which can be efficiently simulated.Research on Quantum Algorithms at the Institute as long as Quantum In as long as mation in addition to MatterJ. Preskill, L. Schulman, Caltechpreskill@caltech.edu / www.iqim.caltech.edu/

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