# Research Highlights and Key Outcomes

The IQIM encompasses four Major Activities (MAs) enumerated below, but the boundaries between these activities are porous and loosely defined. The MAs reinforce one another, and much of our scientific success has been derived from the interactions among the groups. Here we report on some of our significant research accomplishments for the past year.

# MA-1: Quantum Information

Postdoc David Gosset, with Sergey Bravyi, considered how the resources needed for classical simulation of a quantum circuit scale with the number *t *of T gates (rotations by angle p/4), when all the other gates are Clifford gates [1]. Their classical simulation scheme is surprisingly efficient, and may serve as a verification tool for medium-size quantum computations that are dominated by Clifford gates. Demonstrating the power of their new method, they performed a classical simulation of a hidden shift quantum algorithm with 40 qubits, a few hundred Clifford gates, and nearly 50 T-gates. See Figure 1.

**1**. Sergey Bravyi and David Gosset (2016) *Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates.* Physical Review Letters, 116 (25). Art. No. 250501. ISSN 0031-9007. http://resolver.caltech.edu/CaltechAUTHORS:20160620-110726584

**2.** E. Crosson and A. W. Harrow, Simulated quantum annealing can be exponentially faster than classical simulated annealing, http://arxiv.org/abs/1601.03030

For any decision problem, there exists a span program leading to an algorithm with optimal quantum query complexity, but finding such an algorithm is generally challenging. Stacey Jeffery, with Tsuyoshi Ito, formulated new ways to design quantum algorithms using span programs [3]. For example, using their techniques, the span program for OR, which can be used to design an optimal algorithm for the OR function, can also be used to design optimal algorithms for threshold functions (in which we want to decide if the Hamming weight of a string is above a threshold or far below, given the promise that one of these is true), and approximate counting (in which we want to estimate the Hamming weight of the input).

**3.** T. Ito and Stacey Jeffery (2015) *Approximate span programs*, https://arxiv.org/abs/1507.00432 (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20151027-093446138

Extending their previous results, Thomas Vidick and collaborators showed that for a broad class of models (including gapped Hamiltonians with polynomial ground state degeneracy and even some classes of critical systems) ground states of one-dimensional local Hamiltonians can be found efficiently [4]. Their new algorithms bear a close resemblance to renormalization procedures, but in contrast to all previously proposed algorithms the convergence rate can be rigorously bounded. In [5] they improved and simplified one of the main tools used in [4], the so-called detectability lemma. Using the improved lemma they showed that any frustration-free gapped Hamiltonian with spectral gap ϒ can be coarse-grained, to within a length scale of order the inverse square root of ϒ, to yield a local Hamiltonian with constant spectral gap.

**4.** Itai Arad, Zeph Landau, Umesh Vazirani, and Thomas Vidick (2016), *Rigorous RG algorithms and area laws for low energy eigenstates in 1D*, http://arxiv.org/abs/1602.08828 (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20160321-072746620

**5.** Anurag Anshu, Itai Arad., and Thomas Vidick, (2016) *A simple proof of the detectability lemma and spectral gap amplification**.* Physical Review B, 93 (20). Art. No. 205142. ISSN 1098-0121. http://resolver.caltech.edu/CaltechAUTHORS:20160318-153303794

# MA-2: Quantum Matter Theory

Experiments have evolved to the point where we may soon be passing from the `Majorana detection era’ to the `Majorana control era’. Graduate student Dave Aasen, working with postdoc Ryan Mishmash, Jason Alicea and collaborators in Copenhagen, introduced a new all-electrical initialization, manipulation, and readout scheme for Majorana modes in semiconducting-wire devices that leverages tools commonly used in quantum dot and spin qubit studies. They also introduced a series of milestone experiments that reveal foundational properties of Majorana modes directly relevant for quantum computing in the near future, see Figure 2. This work combined the expertise of several members of the IQIM and wouldn’t have been possible without it.

##### 1. Aasen, David and Hell, Michael and Mishmash, Ryan V. and Higginbotham, Andrew and Danon, Jeroen and Leijnse, Martin and Jespersen, Thomas S. and Folk, Joshua A. and Marcus, Charles M. and Flensberg, Karsten and Alicea, Jason (2015) *Milestones toward Majorana-based quantum computing.*(Submitted) http://arxiv.org/abs/1511.05153, http://resolver.caltech.edu/CaltechAUTHORS:20160329-103533737

##### 2. Karzig, Torsten and Oreg, Yuval and Refael, Gil and Freedman, Michael H. (2015) A geometric protocol for a robust Majorana magic gate. (Submitted) https://arxiv.org/abs/1511.05161 http://resolver.caltech.edu/CaltechAUTHORS:20160602-064512689

IQIM postdoc David Mross working with Lesik Motrunich and Jason Alicea provided an explicit derivation of duality between one free Dirac cone and Quantum Electrodynamics in (2+1) dimensions (QED3), see Figure 1. This duality underlies a recent exciting conjecture by Son that composite fermions in the half-filled Landau level have Dirac character [4]. This derivation is essentially constructive and proceeds by coupling (1+1)D wires utilizing bosonization techniques and is well-controlled. In particular, they were able to make novel non-perturbative predictions for the presence and properties of a critical phase in the QED3 field theory. The same constructive approach has already yielded other interesting results – in a very recent paper Mross, Alicea and Motrunich [5] (submitted to PRL), propose an interesting bosonic analog of the Dirac composite fermi liquid.

##### 3. Mross, David F. and Alicea, Jason and Motrunich, Olexei I. (2016) *Explicit derivation of duality between a free Dirac cone and quantum electrodynamics in (2+1) dimensions.* Physical Review Letters, 117 (1). Art. No. 016802. ISSN 0031-9007. http://resolver.caltech.edu/CaltechAUTHORS:20160404-083319765

##### 4. Son, Dam Thanh (2015) *Is the Composite Fermion a Dirac Particle?* Phys. Rev. X 5, 031027

##### 5. Mross, David F. and Alicea, Jason and Motrunich, Olexei I. (2016) *Bosonic Analogue of Dirac Composite Fermi Liquid.* Physical Review Letters, 117 (13). Art. No. 136802. ISSN 0031-9007. http://resolver.caltech.edu/CaltechAUTHORS:20160518-123950203

Symmetry fractionalization (SF) describes the fascinating phenomena that excitations in a topological system can transform under symmetry in a fractional way. For example, it is well known that in fractional quantum Hall systems, excitations can carry fractional charges while the electrons making up the system have charge one. An important realization in the study of SF is that some seemingly consistent SF patterns are actually anomalous and cannot be realized on their own. Xie Chen, with collaborator Michael Hermele, proposed a flux fusion method to detect the subtle difference between anomalous and non-anomalous SF patterns. This analysis provides deeper insight into the physics of 2D spin liquids and 3D symmetry enriched topological phases.

##### 6. Hermele, Michael and Chen, Xie (2015) *Bosonic topological crystalline insulators and anomalous symmetry fractionalization via the flux-fusion anomaly test.* http://arxiv.org/abs/1508.00573 , submitted to Phys. Rev. X http://resolver.caltech.edu/CaltechAUTHORS:20160623-123437341

##### 7. Chen, Xie and Hermele, Michael (2016) *Symmetry fractionalization and anomaly detection in three-dimensional topological phases.* . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20160510-094301838

# MA-2: Quantum Matter Experiment

David Hsieh’s group uncovered the broken symmetries inside the pseudogap region of the high critical temperature (*T*_{c}) superconducting material YBa_{2}Cu_{3}O_{y}. Understanding the nature of the pseudogap region is widely considered to be essential for revealing the mechanism of high-*T*_{c} superconductivity in the copper oxides. Within the last few years, experiments have suggested that the pseudogap is a bona fide thermodynamic phase of matter with distinct broken symmetries. However, exactly what set of symmetries are broken, and in turn what microscopic ordering phenomenon is occurring, remain unsettled questions. Moreover, the fate of this underlying order upon transitioning from the pseudogap region into the superconducting dome is unknown. Using a second harmonic optical rotational anisotropy technique developed in Hsieh’s lab (see Fig. 3), they discovered that global inversion symmetry and all rotational symmetries of the crystal are broken across the pseudogap phase boundary. Moreover, they found that the order parameter of this pseudogap phase persists inside the superconducting dome and does not compete with either the superconducting or charge density wave order parameters. Taken together, their results drastically narrow down the list of possible candidates for the microscopic phase underlying the pseudogap region, favoring those with an odd-parity magnetic order parameter that do not arise from a competing Fermi surface instability.

**Zhao, L. and Belvin, C. A. and Liang, R. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh (2016) ***A global inversion-symmetry-broken phase inside the pseudogap region of YBa_2Cu_3O_y.* Nature Physics . ISSN 1745-2473. (In Press) http://resolver.caltech.edu/CaltechAUTHORS:20161014-080142438

*A global inversion-symmetry-broken phase inside the pseudogap region of YBa_2Cu_3O_y.*Nature Physics . ISSN 1745-2473. (In Press) http://resolver.caltech.edu/CaltechAUTHORS:20161014-080142438

*valleytronics*when combined with modern nanoscale fabrication technology, to produce strained graphene. The strain can induce giant pseudo-magnetic fields (or equivalently, giant

*Berry connections*) that couple to the two inequivalent valleys with opposite signs, which leads to valley polarization under an applied electric field. Thus, controlled strain on perfect graphene can be tailored to yielding the desirable electronic properties for valleytronic applications.

**2.** Yeh, N.-C. and Hsu, C.-C. and Teague, M. L. et al. (2016) *Nanoscale strain engineering of graphene and graphene-based devices.* Acta Mechanica Sinica . pp. 1-13. ISSN 0567-7718. (In Press) http://resolver.caltech.edu/CaltechAUTHORS:20160217-102636750

Tunneling spectroscopy provides a tool to probe the density of states of various electronic materials. IQIM postdoctoral scholar Chandni U working with Jim Eisenstein and collaborators, explored vertical tunneling transport in van der Waals heterostructures with hexagonal boron nitride (hBN) as a tunnel barrier between graphene (or graphite) and a metal (Cr/Au) electrode. They observed a strong suppression of tunneling at low biases, with a gap of about 130 meV in the dI/dV spectrum for metal-hBN-graphene (or graphite) structures. In the graphene devices, the finite zero-bias tunnel resistance was found to depend on the electron density of the graphene layer, while the gap remained unaffected. They also tested graphite-hBN-graphite junctions, in which the strong suppression of tunneling at low energies was found to be absent. See Figure 2.

##### 3. Chandni, U. and Watanabe, K. and Taniguchi, T. et al. (2015) Evidence for defect-mediated tunneling in hexagonal boron nitride-based junctions. Nano Letters, 15 (11). pp. 7329-7333. ISSN 1530-6984. http://resolver.caltech.edu/CaltechAUTHORS:20151015-094529168

# MA-3: Quantum Optics

**1.** J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, *Atom-atom interactions around the band edge of a photonic crystal waveguide,* Proceedings of the US National Academy of Sciences (PNAS) (2016); available as arXiv:1603.02771v1.

**2.** A. Asenjo-Garcia, J. D. Hood, D. E. Chang, and H. J. Kimble, *Atom-light interactions in quasi-1D nanostructures: a Green’s function perspective, *New Journal of Physics (in review, 2016); available at http://arxiv.org/abs/1606.04977.

Numerous bulk crystalline materials exhibit attractive nonlinear and luminescent properties for classical and quantum optical applications. For instance, erbium dopants in crystals exhibit highly coherent optical transitions well suited for solid-state optical quantum memories operating in the telecom band. A chip-scale platform for high quality factor optical nanocavities in these materials will enable new optoelectronic devices and quantum light-matter interfaces. The Faraon group has recently [3, 4] been able to create photonic crystal nanobeam resonators fabricated using focused ion beam milling in bulk insulators, such as rare-earth doped yttrium orthosilicate and yttrium vanadate (see Figure 3). Operation in the visible, near infrared, and telecom wavelengths with quality factors up to 27,000 and optical mode volumes close to one cubic wavelength were demonstrated. Critically, the Faraon group showed coupling of erbium ions to the modes of a photonic crystal cavity, which led to a measured reduction of the photoluminescence lifetime and enhancement of the optical depth in microns-long devices. Such advancements will enable on-chip quantum memories utilizing these erbium doped cavities.

##### 3. Zhong, Tian and Rochman, Jake and Kindem, Jonathan M. and Miyazono, Evan and Faraon, Andrei (2016) High quality factor nanophotonic resonators in bulk rare-earth doped crystals. Optics Express, 24 (1). pp. 536-544. ISSN 1094-4087. http://resolver.caltech.edu/CaltechAUTHORS:20160204-130436067

##### 4. Miyazono, Evan and Zhong, Tian and Craiciu, Ioana et al. (2016) Coupling of erbium dopants to yttrium orthosilicate photonic crystal cavities for on-chip optical quantum memories. Applied Physics Letters, 108 (1). Art. No. 011111. ISSN 0003-6951. http://resolver.caltech.edu/CaltechAUTHORS:20160119-093504030

# MA-4: Mechanical Quantum Systems

The Schwab group has demonstrated “squeezing” of the motion of a micrometer-sized mechanical system (see Figure 4). By driving up the fluctuations in one of the quadratures of the system, they were able to squeeze the fluctuations in the other quadrature. Using the usually feeble force associated with radiation pressure from a microwave source connected to a capacitor on the mechanical system, Schwab and his team manipulated the thermal fluctuations of a micrometer-scale mechanical resonator to produce a stationary quadrature-squeezed state with a minimum variance of 0.80 times that of the ground state uncertainty given by the quantum zero-point motion amplitude. They also performed phase-sensitive, back-action evading measurements of a thermal state squeezed to 1.09 times the zero-point level. These results are relevant to the quantum engineering of states of matter at large length scales, the study of decoherence of large quantum systems, and for the realization of ultrasensitive sensing of force and motion.

##### 1. Wollman, E. E. and Lei, C. U. and Weinstein, A. J. and Suh, J. and Kronwald, A. and Marquardt, F. and Clerk, A. A. and Schwab, K. C. (2015) Quantum squeezing of motion in a mechanical resonator. Science, 349 (6251). pp. 952-955. ISSN 0036-8075. http://resolver.caltech.edu/CaltechAUTHORS:20150728-095253969

So far most experimental achievements in the field of cavity-optomechanics have been realized in setups comprising one optical mode coupled to one vibrational mode. One of the next frontiers is the combination of many such optomechanical cells into an optomechanical array, enabling the optical in situ investigation of (quantum) many-body dynamics of interacting photons and phonons. In collaboration with theorists at the University of Erlangen, the Painter group has analyzed in a new paper [2] how the optomechanical interaction between photons and mechanical vibrations can be used to create artificial magnetic fields for photons on a lattice. In two related experimental efforts [3, 4] the Painter group has also shown just how feasible such experiments may be in the near future. In the first experiment [3], Painter and his team created for the first time a cavity-optomechanical circuit consisting of multiply connected cavity elements via photon and phonon waveguides on a silicon microchip (see Figure 4). In the second experiment, individual optomechanical crystal cavities were measured at milliKelvin temperatures using optical probing techniques and single photon counting. These measurements showed that such optomechanical devices could be cooled deep in the quantum regime (thermal occupany) with extremely long thermal decoherence times approaching a millisecond.