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

Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech [email protected] / 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 Magic state distillation with low overhead. computation. Objective Approach Status • Quantum algorithms for simulating local quantum • Quantum circuit obfuscation schemes based on the systems. connections between quantum circuits and braids. • Novel applications of the quantum Fourier • Proposed quantum-resistant cryptosystem based on transform and other transforms. hardness of solving systems of quadratic equations. • Customizing quantum fault tolerance for physically • Efficient magic-state distillation protocol using a motivated noise models. new class of triorthogonal quantum codes. • Schemes for physically robust quantum storage and • Scheme for performing protected quantum gates processing. based on a continuous-variable quantum codes. • Characterizing Hamiltonian complexity. • Sufficient condition on noise correlations for • Quantum-resistant classical cryptography. scalable quantum computing. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech [email protected] / 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. Research on Quantum Algorithms at the IQIM Faculty: Students: John Preskill Michael Beverland Alexei Kitaev Peter Brooks → HRL Leonard Schulman Bill Fefferman Gil Refael Jeongwan Haah → MIT Faculty Associates: Isaac Kim → Perimeter Todd Brun Alex Kubica Daniel Lidar Shaun Maguire Steven van Enk Kirill Shtengel Sandy Irani Evgeny Mozgunov Sujeet Shukla Postdocs: Undergrads (4 in 2012, 3 in 2013) Gorjan Alagic Nate Lindner Visitors: Glen Evenbly → Technion Many Alexey Gorshkov Spiros Michalakis → NIST Fernando Pastawski Postdocs arriving 2013-14: Zhengcheng Gu Ling Wang Mario Berta (ETH) → Perimeter Beni Yoshida Andrew Essin (Colorado) O. Landon-Cardinal (Sherbrooke) Kristan Temme (MIT) 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, Kitaev honeycomb 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. 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? 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 nb qubits, where nb scales logarithmically with energy and accuracy. Procedure: (1) Prepare free field vacuum. (2) Prepare free field wavepackets. (3) Adiabatically turn on the coupling constant l(t). (4) Evolve for time T using interacting Hamiltonian H. (5) Adiabatically turn off coupling (6) Measure field modes of free theory. Need to discretize the problem, and keep track of resulting errors. Simulating fermionic 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. This year’s work : Consider a self-coupled fermionic field in d = 1 spatial dimensions (e.g., Gross-Neveu model). Procedure: (1) Prepare uncoupled fermion modes. (2) Adiabatically turn on nearest neighbor coupling between modes. (3) Adiabatically turn on the coupling constant l(t). (4) Excite spatially localized wave packets with time-dependent sources. (5) Measure charge and postselect on detecting one particle. (6) Evolve for time T using interacting Hamiltonian H. (7) Nondestructively measure energy and momentum of outgoing particles. Need to discretize the problem, and keep track of resulting errors. Simulating fermionic quantum field theory Jordan, Lee, Preskill 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. Simulating fermionic quantum field theory Jordan, Lee, Preskill Procedure: (1) Prepare uncoupled fermion modes. (2) Adiabatically turn on nearest neighbor coupling between modes. (3) Adiabatically turn on the coupling constant l(t). (4) Excite spatially localized wave packets with time-dependent sources. (5) Measure charge and postselect on detecting one particle. (6) Evolve for time T using interacting Hamiltonian H. (7) 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 4 T= O (1/ a ε) where a is the lattice spacing and e is the error. Using a high G= O( TL / a ) 1+o (1) -order Trotter approximation, the number of gates needed is: ( )

The error due to nonzero lattice spacing scales as e ~ a, hence cost scales with error as 6+o (1) G= O ((1/ε ) ) (seems pessimistic) Simulating quantum field theory Future plans:

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? G. Alagic, Obfuscation T. Jeffery, Take a circuit C and produce another circuit O(C), so that: S. Jordan 1. functionality is preserved; 2. size is not much bigger (say polynomial); 3. 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, Goldwasser Rothblum ’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? G. Alagic, Quantum Obfuscation T. Jeffery, S. Jordan 1. What if we ask for a slightly weaker condition (3)? 2. 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 quantum circuits braids 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. G. Alagic, Approximation theory on groups A. Russell, L. Schulman

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-limited f (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” 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 (Multivariate Quadratic + Code)-Based Cryptosystem

Post-Quantum: honest players are classical and 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 for 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 cryptography

Number 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 for concern: -- Single-register coset measurements info. theoretically sufficient. -- Kuperberg algorithm: Time 2O( n ) McEliece : based on hardness of decoding general linear EC codes. Public key C = M G P G generates linear code, M is random matrix, P is random permutation. Encode v as vC + correctable errors (weight § t). Code is efficiently decodable, and has to be carefully chosen.

New proposal : a code-based scheme in which the scrambled code is not public . Schulman (Multivariate Quadratic + Code)-Based Cryptosystem

Public: three-index 2N µ 2N µ L binary tensor Tijl=∑ k RC ijk kl + ab il jl R is random 2N µ 2N µ K binary tensor , {a, b} are 2L random length-(2N ) vectors, and C is generator of a scrambled length-L efficiently decodable linear code, which can correct most errors of weight r for some r > 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.

  (code) (error) yl= ∑ ij( xrxs )( i ) j∑ RCab ijkkl+ iljl  = y l + y l k  (error) where yl )=1 with probability ¼ (product of two random bits). (error) (error) Decryption : Decode to find yl ). Each l for which yl )=1 provides linear equations for xr and 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 random 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), and then minimum distance decoding is feasible (but still no known codes). With an 7-index tensor the correctable error rate becomes 1/64, and 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 Feigel’man & Ioffe Protected superconducting qubit Doucot & Vidal Kitaev

Physically robust encodings have been proposed using superconducting circuits containing Josephson junctions, for example the “0-Pi qubit”. The circuit’s energy E( q), as a function of the superconducting phase difference q between its leads, is a periodic function with period p to an excellent approximation. “0-Pi qubit”:

0 q

Ef≈(2θ ) + O( exp( − c (size) ))

Two states localized near q=0 and q=p are the basis states of a protected qubit. The barrier is high enough to suppress bit flips, and the stable degeneracy suppresses phase errors. Protection arises because the encoding of quantum information is highly nonlocal, and splitting of degeneracy scales exponentially with size of the device.

Brooks, Kitaev, Preskill Brooks, Protected phase gate Kitaev, j Preskill q π  2 exp i Z  C L 0-Pi qubit LC e k 4  / /(2)≈ 1 Ω 0

For 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 (Clifford group) phase gates can be applied to 0-Pi qubits by turning on and off the coupling between a qubit (or pair of qubits) and 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 j q π  2 exp i Z  C L 0-Pi qubit LC e k 4  / /(2)≈ 1 Ω 0 Under suitable adiabaticity Switch is really a tunable Josephson junction: conditions, closing the switch Q2ϕ 2 transforms a broad oscillator H=+− J( t )cos ()ϕ θ − state (e.g. the ground state) 2C 2 L into a grid state (approximate V( ϕ) codeword).

ϕ ∆ ϕ κ−1 Peaks are at even or odd multiples of π depending on whether θ is 0 or π, i.e. on whether qubit is 0 or 1. Inner width squared is (JC) -1/2 and outer width is (L/C) 1/2 −1 − 1 ω ω J =CJ/ switching time = LC 1 1/2 ()L/ C = 80 ()JC 1/2 = 8

τ J / C = 80

|+〉C Perror (ε )

calculable contribution to error due to diabatic effects and Q-space spreading Large inductance

The intrinsic error scales like exp( -(1/4)L / C ) . Is L / C ≈ 80 reasonable? ϕ

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. What about universal quantum computation and measurement? -- If we can prepare and measure in the basis |0 〉 ± |1 〉, a noisy π/4 single-qubit phase gate (F > .93), augmented by state distillation, suffices for 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! Bravyi, Haah

Magic State Distillation with Improved Overhead

-- In typical protocols for fault-tolerant quantum computing based on stabilizer codes, Clifford operations (e.g. CNOT gates and 90 degree single-qubit rotations) have relatively low overhead cost. -- Overhead tends to be dominated by non-Clifford 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-Clifford gates with error rate e, the cost scales like log g(1/ e), and we would like to reduce the exponent g. -- Exponent is g = log(r) / log(a) if protocol yields one output copy for each r input copies, and reduces error from p to O(pa). -- New family of protocols asymptotically achieves g = log(3) / log(2) ~ 1.6. Best previous protocol had been g = log(5) / log(2) ~ 2.3. Triorthogonal matrix

• A binary matrix where any pair of rows has even overlap and 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 distillation By a stabilizer code based on “triorthogonal matrices” Encoder Decoder Postselect 0-synd. Designed Clifford Designed X-syndrome Meas. X-syndrome

Noisy magic states are represented by a stochastic application of Z-rotation

Distills Pi/8-rotation magic state Distillation cost improved

• Using a new explicit family of triorthogonal matrices G(k), • Error rate improves as • Avg. # of input states to reach a target error rate is

• Numerical optimization, combining various protocols. -12 At Target error rate 10 : C -- 2-fold improvement from Meier-Eastin-Knill (1204.4221) 1000 -- 10-fold improvement from (original) Bravyi-Kitaev (2004) 100 In plot, upper curve is Meier et al. 10 Lower curve is new protocol.

∆ 5 10 15 20 25 30 Haah Entanglement Renormalization

• Local unitary transformation, – On nearest neighbors • factoring out trivial degrees of freedom. • Understand “long-range entanglement” Laurent polynomial representation of stabilizer code Hamiltonians

• Local unitary = row operation • Trivial qubit = presence of sole 1 in a column • Coarse-graining = matrix expansion • Toric code maps to self in a coarse-graining step Cubic code

• Cubic code factorizes to itself plus another.

The other factorizes Coarse-graining step: into two copies of itself. A ö A + B (X-type stabilizers are B ö B + B shown.) Branching MERA Evenbly, Vidal (1210.1895) • Several branches in MERA • Proposed to described highly entangled critical systems. • Cubic code, a gapped spin model, turns out to fit. • Area law holds still, for being in 3D. Highly entangled quantum circuits (arXiv:1210.1895)

t 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ψ0

1 Ψ1

Block entanglement entropy scaling 2 Ψ 2 ≈

3 Ψ scales as the bulk 3 of the block!

T ΨT ≡ Ψ L Minimal Updates in Holography (arXiv:1307.0831)

(D+1) – dimensional holographic local change in description of its ground state Hamiltonian H IR H

z ψ D – dimensional Hamiltonian localized change in UV holographic description H of ground state Minimal Updates ψ ψ new ground state If we use a matrix-product state description, then a local ψ change in the Hamiltonian may require us to modify tensors far away. With a modified holographic description, we Hamiltonian need only modify the tensors within a causal code of H= H + H R bounded width. R Evenbly Tensor network states for disordered systems

Evenbly Kitaev

An area law and sub-exponential algorithm for 1D systems (Arad, Kitaev, Landau, Varzirani). Entanglement entropy of gapped 1D system scales linearly with reciprocol of spectal gap. An algorithm for approximating the ground state which runs in subexponential time.

Finding the group of units in algebraic number rings of arbitrary degree (in progress, Eisentraeger, Hallgren, and Kitaev). Toward a uniformly polynomial algorithm that finds the period of a function on G = Rq for any q.

Classifying locally definable quantum phases of matter (Kitaev). A definition of quantum phases with short-range entanglement, and a proposed topological classification of all such phases in any dimension. Research on Quantum Algorithms at the Institute for Quantum Information J. Preskill, A. Kitaev, L. Schulman, Caltech [email protected] / www.iqi.caltech.edu/

• Progress on last year’s objectives – FY11-12 - Quantum algorithms for simulating particle collisions in strongly-coupled quantum field theories. - Proposed trap-door one-way functions based on tensor problems. - Quantum algorithms for approximating invariants of triangulated manifolds by tensor contraction. - Classical certificates for frustration-free ground states of commuting Hamiltonians on square lattices. - Estimating fidelity using a number of Pauli operator expectation values independent of system size. - Enhanced memory time for three-dimensional quantum memories without string operators. - Performance analysis for fault-tolerant quantum computing based on asymmetric Bacon-Shor codes. - Nonlocal order parameters for symmetry-protected phases in one dimension. - Studies of resonating valence bond (RVB) states using the PEPS formalism. • Research plan for the next 12 months – FY12-13 - Extend algorithms for simulating quantum field theories to fermions, gauge fields, massless particles. - Quantum algorithms for preparing the Gibbs states of local quantum systems at nonzero temperature. - Develop efficient band-limited quantum Fourier transforms over Lie groups. - Quantum circuit obfuscation schemes based on the connections between quantum circuits and braids. - Near-optimal dynamical decoupling schemes for multi-level quantum systems. - Classification of phases for “locally definable” quantum systems in arbitrary dimensions. • 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. Research on Quantum Algorithms at the Institute for Quantum Information and Matter J. Preskill, L. Schulman, Caltech [email protected] / www.iqim.caltech.edu/