Research on Quantum Algorithms at the Institute for Quantum Information and Matter

1. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech preskill@caltech.edu / www.iqim.caltech.edu/ Objective • Improved rigorous estimates of thresholds for fault-tolerant quantum computation. • Quantum algorithms beyond the hidden subgroup paradigm. • Quantum and classical simulation methods for quantum many-body systems. • New approaches to physically robust quantum computation. Dissipative preparation of topological states. Objective Approach Status • Quantum algorithms for simulating local quantum systems. • Novel applications of the quantum Fourier transform and other transforms. • Customizing quantum fault tolerance for physically motivated noise models. • Schemes for physically robust quantum storage and processing. • Characterizing Hamiltonian complexity. • Quantum-resistant classical cryptography. • Quantum algorithms for simulating particle collisions in fermionic quantum field theories. • Optimal algorithms for dissipative preparation of topological code states. • Characterization of fault-tolerant logical operations in topological subsystem codes. • Proof of rapid thermal mixing for graph state Hamiltonians and free fermions. • Stable, geometry-preserving discretizations of arbitrary compact groups.

2. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech preskill@caltech.edu / www.iqim.caltech.edu/ • Progress on last year’s objectives – FY13-14 • Quantum algorithms for simulating particle collisions in fermionic quantum field theories. • Results on optimal contractions of generic tensor networks, with applications to critical systems. • Optimal algorithms for dissipative preparation of topological code states. • Characterization of fault-tolerant logical operations in topological subsystem codes. • Evidence for probability distributions that can be sampled efficiently quantumly but not classically. • Asymptotically uniform, stable, geometry-preserving discretizations of arbitrary compact groups. • Properties of alpha-Renyi generalizations of quantum conditional mutual information. • New framework for quantum-proof pseudorandomness based on operator space theory. • Proof of rapid thermal mixing for graph state Hamiltonians and free fermions. • Research plan for the next 12 months – FY14-15 • Show that quantum field theory simulation solves a hard (BQP-complete) problem. • Characterize the logical operators that can be performed using constant-depth unitaries in TQFT.s • Stronger impossibility proofs for quantum obfuscation schemes. • Relate quantum-proofness of randomness condensers to Bell inequality violation. • Show that quantum messages and entanglement increase the power of multi-prover interactive proofs. • Improved rigorous bounds on quantum memory times. • Long term objectives • - Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for protecting quantum systems from noise. • - Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.

3. Research on Quantum Algorithms at the IQIM Faculty: John Preskill Alexei Kitaev Leonard Schulman Gil Refael Thomas Vidick Faculty Associates: Todd Brun Steven van Enk Sandy Irani Postdocs: GorjanAlagic → Copenhagen Mario Berta Andrew Essin Glen Evenbly → UC Irvine Olivier Landon-Cardinal Students: Michael Beverland Bill Fefferman→ U. Maryland Matt Fishman Alex Kubica Shaun MaguireEvgenyMozgunov Sujeet Shukla Nicole Yunger-Halpern Undergrads (6 in 2014) Visitors: Many Postdocs arriving 2014-15: Omar Fawzi (ETH) David Gosset (Waterloo) Stacey Jeffery (Waterloo) Daniel Lidar Kirill Shtengel SpirosMichalakis Fernando Pastawski KristanTemme Ling Wang Beni Yoshida

4. Simulating quantum evolution -- Goal: simulating real-time evolution of many-particle systems with better accuracy than is classically achievable. -- Simulating quantum chemistry. -- Simulating quantum matter. -- Simulating quantum field theory. -- Simulating quantum gravity. -- AdS/CFT, BFSS Matrix model, etc. -- Thinking about how to simulate quantum field theory on a classical computer elucidated the foundations of field theory. Will thinking about how to simulate string theory on a quantum computer help us answer: What is string theory?

5. Jordan, Lee, Preskill Quantum algorithms for quantum field theories -- Feynman diagrams have limited precision, particularly at strong coupling. -- Classical lattice methods can compute static properties, but cannot simulate dynamics A quantum computer can simulate particle collisions, even at high energy and strong coupling, using resources (number of qubits and gates) scaling polynomially with precision, energy, and number of particles. -- Estimate errors due to regulating (spatial lattice and approximating continuous variable fields by qubits). -- Efficient procedure for preparing (strongly-coupled) vacuum and initial wave packet states, simulating time evolution, measuring final state. Does the quantum circuit model capture the computational power of Nature? What about quantum gravity?

6. Simulating quantum field theory Jordan, Lee, Preskill Input: 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 nbqubits, where nb scales logarithmically with energy and accuracy. Procedure: Prepare free field vacuum. Prepare free field wavepackets. Adiabatically turn on the coupling constant (t). Evolve for time T using interacting Hamiltonian H. Adiabatically turn off coupling Measure field modes of free theory. Need to discretize the problem, and keep track of resulting errors.

7. Jordan, Lee, Preskill Simulating quantum field theory Example: self-coupled scalar field in d = 2 spatial dimensions. Scaling with error dominated by preparation of (free) vacuum: Scaling with energy: Factor of E from Trotter step, E2 from a ~ E-1, E3from diabatic effect. Need to discretize the problem, and keep track of resulting errors. Errors arise from, e.g. -- digitization of field. -- lattice spacing a. Error is O(a2). -- Trotter step size. -- diabatic particle production during state preparation. Formally, an infinite number of degrees of freedom per unit volume. But good approximation with (energy-dependent) lattice spacing. Even for 2  4 scattering in 1 spatial dimension, ~10,000 qubits for 1% accuracy. Can improve E dependence from diabatic particle production somewhat… How much can we improve scaling with error epsilon? Requires a lattice theory with a larger lattice spacing a in physical units(RG improvement.)

8. Jordan, Lee, Preskill arXiv:1404.7115 Simulating fermionic quantum field theory Input: 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. Completed this year (in progress last year): 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 and postselect on detecting one particle. Evolve for time T using interacting Hamiltonian H. Nondestructively measure energy and momentum of outgoing particles. Need to discretize the problem, and keep track of resulting errors.

9. Jordan, Lee, Preskill arXiv:1404.7115 Simulating fermionic quantum field theory Free 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> and |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) and can be updated in time O(log L). Exciting wave packets: Modulate source spatially and temporally to match one particle states. Make the source weak to avoid creating more than one particle, but it usually produces nothing. Measure and abort if not particle created (okay for a collision of a constant number of particles). Advantage over previous method (in which coupling ramps on after wavepacket created): works for bound states.

10. Simulating fermionic quantum field theory Jordan, Lee, Preskill 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 and postselect on detecting one particle. Evolve for time T using interacting Hamiltonian H. Nondestructively measure energy and momentum of outgoing particles. Need to discretize the problem, and keep track of resulting errors. Cost is dominated by the adiabatic preparation of the vacuum. Adiabaticity enforces turn-on time where a is the lattice spacing and  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, and another factor of 1/  arises from postselection; hence cost scales with error as (seems pessimistic!)

11. Simulating quantum field theory Future plans: BQP-completeness of simulating quantum field theory. Improve cost. Massless particles (infrared safe observables). Gauge fields (start with strong coupling limit). Ground state preparation by cooling. Nonzero temperature and chemical potential. Simulate standard model of particle physics in BQP. Quantum gravity?

12. BQP-hardness of simulating quantum field theory (in progress) -- Is simulating an interacting quantum field theory as hard as any problem a quantum computer can solve efficiently? -- Formulate the problem in terms of the response of the vacuum to smooth spatially and temporally modulated source fields. -- Adiabatically turn on and populate potential wells, for a “dual-rail” encoding of qubits. -- Perform a universal set of quantum gates using phase shifts due to particle interactions and tunneling between wells. -- Avoid measurements until the end of the protocol (with adaptive number operator measurements even bosonic free fields can simulate a universal quantum computer). Rules out quantum error correction. -- Check that it works for a self-coupled massive scalar field.

13. Simulating quantum field theory Future plans: BQP-completeness of simulating quantum field theory. Improve cost. Massless particles (infrared safe observables). Gauge fields (start with strong coupling limit). Ground state preparation by cooling. Nonzero temperature and chemical potential. Simulate standard model of particle physics in BQP. Quantum gravity?

14. Quantum obfuscation An obfuscator O is an algorithm that maps circuits to circuits, and • preserves functionality: C and O(C) implement same function; • preserves efficiency: |O(C)| < poly(|C|); • obfuscates: any attack A on O(C) has a corresponding black-box simulator SC . Idea: the circuit O(C) is no more useful than a black box with same functionality. What if we had one? Generically… • provable software intellectual property protection; • software patches that don’t reveal the security hole being patched; • turn any private-key encryption to public-key encryption (public key is O(Enck)); • delegated computation; • etc. Unfortunately… no-go: Barak et al. 2001 showed (black-box) obfuscation is impossible classically.

15. Quantum obfuscation An obfuscator O is an algorithm that maps circuits to circuits, and • preserves functionality: C and O(C) implement same function; • preserves efficiency: |O(C)| < poly(|C|); • obfuscates: any attack A on O(C) has a corresponding black-box simulator SC . What if O is a quantum algorithm and can output a quantum circuit, or a state? Work in progress [Alagic & Fefferman]: • proving impossibility when O(C) is a quantum circuit; • proving impossibility of two-circuit-obfuscation when O(C) is a quantum state. Two-circuit obfuscation replaces condition 3 above with: • (2-circuit) any attack A on a pair (O(C), O(D)) has a black-box simulator SC, D. …harder to achieve (thus easier to prove impossible.)

16. Quantum obfuscation Work in progress [Alagic & Fefferman]: • proving impossibility when O(C) is a quantum circuit; • proving impossibility of two-circuit-obfuscation when O(C) is a quantum state. Proof approach similar to original Barak et al. paper: • pick a random string and two unitaries U1, U2 that only differ on that string; • write down equal-length circuits C1, C2 for both; • write down a “universal circuit” D that runs its input on the secret string; • design an adversary that will accept the quantum states O(Cj) and O(D) and run the second one on the first one, thereby discovering j; • show that any black-box simulator faces an unstructured search over exponentially large spaces of strings and circuits. To get a one-circuit proof, show how to patch D and Cj together into a single circuit (seems to require cloning if O produces states!)

17. Quantum obfuscation What about other definitions of obfuscation? Replace 3 with: • (best-possible) O(C) leaks the least information among all circuits that compute fC. 3. (indistinguishability) fC1 = fC2 implies O(C1) = O(C2). Work in progress [Alagic & Fefferman]: • impossibility results for both of the above, when the guarantees in the definition are perfect or statistical; • basic approach: show how such obfuscators would give a mapping from quantum state distinguishability to quantum operation distinguishability; • easy: perfect pure-state obfuscation of classical functions would put coNP in BQP; • harder: any statistical guarantee for unitaries would put coQMA in QSZK; • analogous classical results collapse the polynomial-time hierarchy (unlikely!)

18. Classical simulation of Yang-Baxter What is the Yang-Baxter equation? Recall braid groups… • n pegs on top and bottom; • strands connect top pegs to bottom pegs; • continuous deformations are ok; • forms a group. Simple way to create a unitary representation of this group: • assign a qudit to each gate; • pick a 2-qudit gate R; • assign twist of strands i and i+1 : • R must satisfy Yang-Baxter equation: Why care about local unitary representations of the braid group? Anyons! = 1

19. Classical simulation of Yang-Baxter When kind of quantum-algorithmic power do we get by braiding in this model? [Alagic, Bapat (Caltech undergrad) and Jordan TQC ’14]: • if d = 2, it’s purely classical power; • this is true for some higher-dimensional solutions. Approach: • use a classification of the qubit solutions due to Hietarinta and Dye; • provide classical algorithms for simulating circuits constructed from braiding using such a solution; • show that some of the solution families extend to higher dimensions, and that our algorithm applies to such solution families; • algorithm also applies to other (non-braiding) quantum circuit families not previously known to be classically simulable.

20. Classical simulation of Yang-Baxter When kind of quantum-algorithmic power do we get by braiding in this model? [Alagic, Bapat (Caltech undergrad) and Jordan TQC ’14]: • if d = 2, it’s purely classical power; • this is true for some higher-dimensional solutions. Two categories of solutions: • solutions which are “almost” Clifford gates: these are amenable to an exact stabilizer-style simulation algorithm for observables on at most polylogarithmically-many qubits; • other solutions are a bit more complicated: we used a path-integral approach with probabilistic sampling, and achieve polynomial-time classical calculation of circuit amplitudes with 1/polynomial precision; • potential future work: when do such solutions correspond to anyon models, or yield link invariants? Can we characterize the case d = 3? Is it ever universal?

21. Complexity, quantum, and topology [Alagic] Current project with Caltech undergrad Catharine Lo. Idea: • take a well-established conjecture in classical computational complexity; • apply some tricksfrom quantum algorithms; • get a statement about topology of 3-manifolds. Recall: • a 3-manifold is a space which locally looks like Euclidean 3-space; • gluing solids sometimes gives 3-manifolds: glue two solid balls together and you get the 3-sphere (just like two disks glued along boundary give 2-sphere); • Theorem: can get all3-manifolds just by gluing smooth solids with handles! • there’s a choice: what map to use to do the gluing.

22. Complexity, quantum, and topology Recall: • a 3-manifold is a space which locally looks like Euclidean 3-space; • gluing solids sometimes gives 3-manifolds: glue two solid balls together and you get the 3-sphere (just like two disks glued along boundary give 2-sphere); • Theorem: can get all3-manifolds just by gluing smooth solids with handles! • there’s a choice: what map to use to do the gluing. Further: • the Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by: gluing map, given as a list of geometrically local twists Manifold: glue g-handled solids along map f local unitary representation of the group of all gluing maps

23. Complexity, quantum, and topology The Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by: Idea one: • give a straightforward quantum circuit which encodes the number #h of satisfying assignments of a formula h into a matrix entry of the circuit; • show that exponentially small precision is sufficient to recover #h; • use density of representation Ug and Solovay-Kitaev to show that approximating WRT with exponentially small precision is thus #P-hard. Manifold: glue g-handled solids along map f local unitary representation of the group of all gluing maps gluing map, given as a list of geometrically local twists

24. Complexity, quantum, and topology The Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by: Idea two(inspired by work of M. Freedman): • assume that (standard in classical complexity theory); • Theorem: there exist manifold diagrams Mg, f satisfying: every nearby diagram of same manifold must have genus at least polylog(g). • “nearby” means poly-many equivalence moves (analogous to Reidemeister); • why is this true? Notice: can get WRT exactly when space is of poly(n) dimension. • so if Theorem were false, the list of equivalence moves together with this exact calculation would be a poly-length classical proof for a #P-hard problem! Manifold: glue g-handled solids along map f local unitary representation of the group of all gluing maps gluing map, given as a list of geometrically local twists

25. Approximation theory on groups Recall that the Quantum Fourier Transform (QFT) is the basis of almost all known exponential quantum speedups. Can we find other transforms of this kind? The QFT is a discrete transform. What about continuous spaces? To even get this idea off the ground, we need integration; this in turn requires • a way to discretize the continuous space; • a sampling theory with good numerical stability. There are some results of this kind for specific groups and spaces (e.g., finite symmetric group, SO(3), tori, spheres). Our goal: develop a generic theory for all compact groups. [Alagic Russell Schulman]

26. Approximation theory on groups Results: Theorem 1. Let G be a compact group and B a finite set of irreducible representations of G. For every there is a faithful -uniform quadrature rule for B-band-limited functions of size . What does this mean? • irreducible representations:nonabelian analogue of frequencies; • -uniform quadrature rule: finite set of sample points that allows for exact integration, with weights within of uniform; • B-band-limited function: a function which can be expressed as a linear combination of “frequencies” from B only; Point: polynomial-size nearly-uniform quadrature can be achieved generically on almost any reasonably well-behaved group.

27. Approximation theory on groups Results: Theorem 2. Let (X, w) be an -uniform quadrature rule for End(B) and f, g two B-band-limited functions. Then What does this mean? • End(B) is a larger set than B (it’s the tensor square); • the quadrature that we achieved in Theorem 1 is actually strong enough to give something really nice: • the set of samples is actually a really good geometric proxy for the group; sampling functions approximately preserves inner products. These mathematical results prove that certain algorithms (classical or quantum) on continuous spaces are possible in principle. Designing such algorithms is the next step.

28. Dengis, König,PastawskiNJP (2014), 16, 1, 013023 König,PastawskiPRB (2014) 90, 045101 Dissipative preparation of topological states -- Dissipation is omnipresent in physical systems and provides a more general description to their evolution. -- It can in principle be engineered to drive phase transitions, state engineering or even quantum computation. Can we use dissipation to produce topologically entangled states in simple ways? Do dissipative processes induce a different notion of phases? Anyons are ``swept’’ in to a fixed sink site. Result 1: Encoding into the toric code may be achieved by a locally generated dissipative process which is almost translation invariant in a time proportional to the torus perimeter. Result 2: No locally generated dissipative process may generate topological ordered states from trivial ones in a time sublinear in the code distance (assuming non-trivial topology). Proven using Lieb-obinson bounds and presence of long range correlations.

29. Tradeoffs for protected gates in quantum codes -- Guide the search for frugal realizations of fault-tolerant quantum computation. ( Where we should not push further and why.) -- Understand existing results in the context of general error correcting codes. -- Classical computers may simulate Clifford group operation and Pauli measurements efficiently. -- Magic states + Cliffords enable universal computation( They can be distilled from non-Clifford gates ). -- Geometrically local codes are easier to implement experimentally. -- Geometrically local gates seem a reasonable proxy for fault tolerance. What can we do with TQFTs?Topological quantum field theories What can we do with subsystem codes? Beverland, Pastawski, Sijher, König, Preskill Pastawski, Yoshida

30. Pastawski, Yoshidaarxiv:1408.xxxx Tradeoffs for protected gates in subsystem codes • Result 1: Stabilizer codes locally defined on a three-dimensional lattice may not have both of the following properties: • An associated Hamiltonian with a macroscopic energy • barrier. • A constant depth logical gate outside the Clifford group. • Result 2: The code distance of a D-dimensional stabilizer code ( LD ) with a constant depth implementation of a gate in the m-th level of the Clifford hierarchy is bounded by O(LD+1-m). • Result 3: Subsystem code families with transversal implementation of gates in the m-th level of the Clifford have a qubitloss threshold • upper bounded by 1/m. Result 4: Consider subsystem code families locally defined in a D-dimensional lattice, having a constant loss-threshold and logarithmically growing code distance. Then the set of logical unitaries implementable by constant depth local circuits is included in the D-th level of the Clifford hierarchy. (Use Hastings 2011 to find correctable region)

31. Locality preserving gates in topological quantum field theories Beverland, Pastawski, Sijher, König, Preskill • Hypothesis: A deformation lemma for string-like logical operators is assumed and plays the same role the cleaning lemma does for stabilizer and subsystem codes. • Result 1 (in progress): For rational abelian anyon models, a result fully analogous to the one for the toric code is recovered. • Focus on geometry preservation allows providing a sharper constraint on the set of allowed ``generalized Cliffords’’. • Result 2 (in progress): Characterization of locality preserving logical gates on TQFTs supporting non-abelian anyons. • Fibonacci, Ising, … (more restrictive than for abelian case!) • Local constraints from simple closed loops. • Global constraints from fusion rules. • Global constraint from basis change.

32. Conditional (Quantum) Mutual Information • Measures correlations between A and B from the perspective of C: A B C • Nice properties: • Non-negative (strong subadditivity of quantum entropy) • Monotone under local operations • Self-duality (monogamy of entanglement) • Many features known if C is classical but only poorly understand if C is quantum: • Relation to conditionally independent states (quantum Markov states) or weakly correlated states? • Important quantity in: • Entanglement theory • Quantum coding theory • (Quantum) communication complexity • Condensed matter physics

33. Renyi Conditional Quantum Mutual Information • Berta,Wilde, Seshadreesan(arXiv:1403.6102): introduce continuous Renyi one-parameter family with based on α-Renyi relative entropy. • We show that keeps the same properties as the original conditional quantum mutual information: • Non-negative • Monotone under local operations • Self-duality • So far mostly exploring mathematical properties (heavy matrix analysis). • Possible applications in entanglement theory, quantum coding theory, quantum communication complexity, condensed matter physics. • α-Renyi (relative) entropies are monotone in the parameter α, what about α-Renyi conditional mutual information? (important for applications, proof only for special cases)

34. Quantum-Proof Pseudorandomness Q: (quantum) observer • The (classical) theory of pseudorandomness: • Randomness extractors • Randomness condensers • Expander graphs • List-decoding codes • Samplers, Disperser, etc. Q Q N Ext M • Example: Randomness extractor. N: input with some entropy M: perfectly random output • Manifold applications, quantum-proof especially important in: • Quantum cryptography • (Classical) post-quantum cryptography • Device independent randomness extraction/amplification • Etc. • Study the power of quantum memory (in otherwise fully classical setup). • Highly non-trivial effects, no generic understanding.

35. Generic Understanding of Quantum-Proof • New framework for understanding quantum-proof (Berta,Fawzi, Scholz): • Extractor and condenser property is a bounded norm condition between Banach spaces • Quantum-proof extractor and condenser property is a completely bounded norm condition between operator spaces (non-commutative Banach spaces) • Spectral randomness extractors are always fully quantum-proof (Berta,Fawzi, Scholz, Szehr: arXiv:1402.3279). • Operator space theory crucial for understanding Bell inequalities. • Many tools from operator space theory to explore, preliminary results (ongoing work): • Extractors with high entropy input are always approximately quantum-proof • Large separation between bounded and completely bounden norm for condensers (only for the norms so far) • Condenser property is a special instance of the bipartite densest sub-graph problem (what is quantum-proof densest sub-graph?)

36. Future Goals • Understand mathematical properties of conditional mutual information: give operational description of quantum states with small quantum conditional mutual information. • Explore applications of Renyi conditional mutual information: • strong converses in quantum coding theory • lower bounds in quantum communication complexity • topological order in condensed matter physics • Pseudorandomness: use operator space theory tools to generically understand quantum-proof. • Possible connection to Bell inequalities (two-player games). • Are specific pseudorandomness constructions quantum-proof (e.g., Parvaresh-Vardy codes)?

37. Temme, Pastawski, Kastoryano Hypercontractivity of quantum semi-groups (arXiv: 1403.5224) • The Question: How fast does a (cptp) semigroup “smear out” all information about its initial condition and equilibrate? • -- We analyse this by the Hypercontractivity of the semigroup: • where . . • (the weighted 2 --> p-norm measures how spiked the observable f is and Hypercontractivityindicates that these spikes get smoothed out exponentially fast.) • -- The “differential formulation” of this property is equivalent to a • log-Sobolev inequality • (Here we compare the “Entropy” of an observable with an “Energy like functional” of the Observable. The constant alpha is called the Log-Sobolev constant.) • The Strategy: We show Hypercontractivity for some fixed p=4 and time t*. Then we use Riesz-Thorin / Stein-Weiss interpolation Theorems to obtain bound on the Log.Sobolevconstant . This analysis works well when the generator has “particle like excitations” that are preserved.

38. Temme, Pastawski, Kastoryano Hypercontractivity of quantum semigroups (arXiv: 1403.5224) • So why should we care?: • 1) Exponentially improved (over spectral gap) mixing time bounds useful in • dissipative state preparation, analysis of quantum algorithms and • thermalization time estimates. • 2) Useful weighted Lp-norm inequalities. • 3) Ensures (together with the locality of the generator) stability of the fixed point • under local perturbations. [Cubitt, Lucia, Michalakis, Perez-Garcia] • Results: We prove bounds on the Log-Sobolevconstant for • 1) Product semigroups with generators: , which allows • to estimate 2 ->p norms of product channels: • 2) Thermalizing Davies generators of: • -- Graph state Hamiltonians. This ensures that the cluster-state of N qubits • can be prepared in time O(polylog(N)), by cooling with T = O(1/log(N)). • -- Free fermion Hamiltonians. (Superconducting wires, etc ...)

39. Future directions: • -- Rigorous upper and lower bounds to the quantum memory time: • Common figure of merit: “The energy barrier of logical operators” • 3D codes with EB: -- Haah code m = O(log(N)) • -- Michnicki’s Welded Toric code m = O(L^(2/3)) • But lifetime only scales up to optimal constant value, hence the EB is missing something. • Open questions: • -- Can we find both upper and lower bounds for the lifetime of quantum • memories ? • -- Can one show that the energy barrier is a necessary criterion for a • thermally stable quantum memory ? • -- Can we derive a better criterion from Poincare type inequalities • for thermalizing Davies semigroups ? • Some results in this direction: • i) O(N) thermalization bound in the Dobrushin - Shlosman regime (i.e. high T) • ii) Lower bounds to the spectral gap of Davies semigroups for stabilizer • codes.

40. Improved efficiency in variational tensor network algorithms If the environment of one tensor can be computed with cost If the environment of one tensor can be computed with cost (G.Evenbly, R. Pfeifer, Phys. Rev. B 89, 245118 (2014)) • Several theorems relating to the contraction of generic tensor networks are presented For an arbitrary closed network These results, when incorporated into numerical tensor network libraries, can significantly improve the performance of a wide variety of simulation algorithms e.g. MERA, PEPS + many more cost: cost: cost: Theorem 1: (equivalence of environments) Theorem 2: (multiple environments) Then the environment of any other tensor can be computed with the same cost Then set of all environments can be computed with fixed cost

41. Tensor Network Renormalization Group (G.Evenbly, work in progress) • Efficient coarse-graining algorithms for a large class of D-dimensional tensor networks • Basis for efficient numerical algorithms for simulating classical many-body systems • on D-dimensional lattices or quantum systems on (D-1)-dimensional lattices Insert unitaries and projectors Initial tensor network (e.g. representing the partition function of a 2D classical system) Simplify Coarser tensor network • New approach (TNRG) is effective for critical and gapped systems (whereas previous tensor RG schemes could not properly address critical systems) Contract

42. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech preskill@caltech.edu / www.iqim.caltech.edu/ • Progress on last year’s objectives – FY12-13 • Quantum algorithms for 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 and braids. • Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes. • Efficient algorithm for testing stability of two-dimensional tensor-network states vs. local perturbations. • Scheme for performing protected quantum gates based on a continuous-variable quantum codes. • Sufficient condition on noise correlations for scalable quantum computing. • Near-optimal dynamical decoupling schemes for multi-level quantum systems. • New class of highly entangled many-body states which can be efficiently simulated. • Research plan for the next 12 months – FY13-14 • Quantum algorithms for simulating quantum field theories with gauge fields and massless particles. • Quantum algorithms for simulating thermalization of quantum systems. • Quantum algorithms for interpolating band-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 for strongly disordered systems. • Long term objectives • - Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for protecting quantum systems from noise. • - Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.

43. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech preskill@caltech.edu / www.iqim.caltech.edu/ • Progress on last year’s objectives – FY13-14 • Quantum algorithms for simulating particle collisions in fermionic quantum field theories. • Results on optimal contractions of generic tensor networks, with applications to critical systems. • Optimal algorithms for dissipative preparation of topological code states. • Characterization of fault-tolerant logical operations in topological subsystem codes. • Probability distributions that can be sampled efficiently quantumly but not classically. • Asymptotically uniform, stable, geometry-preserving discretizations of arbitrary compact groups. • Properties of alpha-Renyi generalizations of quantum conditional mutual information. • New framework for quantum-proof pseudorandomness based on operator space theory. • Proof of rapid thermal mixing for graph state Hamiltonians and free fermions. • Research plan for the next 12 months – FY14-15 • Show that quantum field theory simulation solves a hard (BQP-complete) problem. • Characterize the logical operators that can be performed using constant-depth unitaries in TQFT.s • Stronger impossibility proofs for quantum obfuscation schemes. • Relate quantum-proofness of randomness condensers to Bell inequality violation. • Show that quantum messages and entanglement increase the power of multi-prover interactive proofs. • Improved rigorous bounds on quantum memory times. • Long term objectives • - Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for protecting quantum systems from noise. • - Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.

44. 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 for interpolating band-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 for superconducting qubits, universal dynamical decoupling, asymmetric Bacon-Shor codes for protection against biased noise. • Experiment and implementation. Attractive photons in a quantum nonlinear mediua, Kitaevhoneycomb and 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 and sub-exponential algorithm for 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 for strongly disordered systems.

45. G. Alagic, T. Jeffery, S. Jordan Obfuscation Take a circuit C and 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 for all circuits? State of affairs in research • lots of motivation (software/hardware copy protection, homomorphic encryption, turning private key schemes into public key schemes, etc.) • known formalizations of (3) are all too hard: • O(C) no more useful than a black box that performs C? (impossible, Barak et al ’01) • O(C1) indistinguishable from O(C2) for equivalent C1, C2? (collapses PH, GoldwasserRothblum ’07) • little is known about quantum obfuscation • are there classical algorithms for obfuscating quantum circuits? • are there quantum states that allow us to do obfuscated computation?

46. G. Alagic, T. Jeffery, S. Jordan Quantum Obfuscation • What if we ask for a slightly weaker condition (3)? • Can we obfuscate quantum circuits? Results [Alagic Jeffery Jordan ’13] • efficient classical algorithms for obfuscating both quantum and classical circuits • “weaker” condition 3: indistinguishability under a subset of the set of all circuit relations Core idea • if we had an efficient canonical form for circuits (a coNP-hard problem), we would satisfy Goldwasser-Rothblum trivially • but topological quantum computation gives us a pretty good mapping and braids do have efficient canonical forms! • in fact, this mapping exists for classical reversible circuits too, if we use a different representation of the braid group • If 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. quantum circuits braids

47. G. Alagic, A. Russell, L. Schulman Approximation theory on groups The Discrete Fourier Transform (DFT) • basis of countless proofs, algorithms, signal processing tasks, etc. • the fast classical (FFT) algorithms for computing the DFT are very useful in practice • their quantum analogues (QFT) are exponentially faster (in a certain sense) and are a basis for amazing things like Shor’s algorithm What 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 band-limitedf (doesn’t oscillate too much), and sample the function at a nicely spaced finite set of points • for the circle, this boils down to “discretize and use DFT”

48. Approximation theory on groups New 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 case? • signals in practice might be continuous instead of discrete • we care about nonabelian spaces (e.g., spherical harmonics, SU(2)) • we need more quantum-algorithmic primitives for exponential speedups Results [Alagic Russell Schulman 2013] • a theorem about reconstructing band-limited functions on compact groups • setting: any compact group • input: random samples of a band-limited function f • output: the list of Fourier coefficients of f • a number of samples cubic in the band limit is sufficient for a good estimate • the reconstruction is inner-product-preserving in the limit

49. Minimal Updates in Holography (arXiv:1307.0831) (D+1) – dimensional holographic description of its ground state local change in Hamiltonian D – dimensional Hamiltonian Minimal Updates new ground state localised change in holographic description of ground state If we use a matrix-product state description, then a local change in the Hamiltonian may require us to modify tensors far away. With a holographic description, we need only modify the tensors within a causal code of bounded width. modified Hamiltonian Evenbly

50. Tensor network states for disordered systems Evenbly