Research focusing on the foundations of the subject:

1. Quantum Information Science John Preskill DoE Review 24 July 2007 Research focusing on the foundations of the subject: 1) Algorithms/Complexity: Quantum algorithms that achieve speedups relative to classical algorithms, and limits on such algorithms. 2) Entanglement: Quantum entanglement and the theory of transformations among quantum states. 3) Cryptography/Communication: Security of quantum cryptographic protocols, and other types of communication using quantum states. 4) Error Correction/Fault Tolerance/Control: Protection of quantum information, and using quantum error-correcting codes, fault-tolerant protocols, and quantum feedback for quantum information processing. 5) Implementation/Experiment: Physical implementations of quantum information processing. 6) Fundamental physics: Connections between quantum information science and other aspects of fundamental science.

2. 2007 Ph.D.s Panos Aliferis “Level reduction and the quantum threshold theorem”  IBM, Yorktown Heights Parsa Bonderson “Nonabelian anyons and interferometry”  Microsoft Station Q, Santa Barbara Michael Zwolak “Dynamics and simulation of open quantum systems”  Los Alamos National Lab

3. Other Students Ersen Bilgin, Using “belief propagation’’ algorithms for efficient simulation of many-body quantum systems. Kovid Goyal, Fault-tolerant quantum computation based on “resource states” and topological coding. Prabha Mandayam, Formulating necessary and sufficient conditions for approximate quantum error correction. Hui Khoon Ng, Applying the effective field theory method to quantum noise. Greg ver Steeg, Characterizing the computational power of systems with continuous quantum variables, including quantum field theory. Note: My students are not financially supported by DoE (except for Kovid Goyal, the theory group computer system administrator). DoE contributes 9% of my salary.

4. Quantum nonlocality, communication, cryptography (2006-07) • quant-ph/0610203 (Lo, Preskill) Security of quantum key distribution using weak coherent states with nonrandom phases. Security proof that applies when key information is encoded in the relative phase of a coherent-state reference pulse and a weak coherent-state signal pulse. The proof works even if the reference pulse is bright and has a phase known by the adversary. • arXiv:0705.4282 (Blume-Kohout, Ng, Poulin, Viola) The structure of preserved information in quantum processes. Introduced a general characterization of information-preserving structures. Proved that the fixed states and observables of an arbitrary process are linearly isomorphic to a matrix algebra. Found an efficient algorithm for finding all noiseless subsystems. • quant-ph/0611001 (Toner, Verstraete) Monogamy of Bell correlations and Tsirelson’s bound. Study of Bell inequality violation in a tripartite system ABC, characterizing the trade-off between the nonlocality of the Bell correlations observed by AB and of those observed by AC. • quant-ph/0704.2903 (Kempe, Kobayashi, Matsumoto, Toner, Vidickl) On the power of entangled provers: immunizing games agains entanglement. Two generic ways to make multi-prover classical games resistant against entangled provers.

5. Fault-tolerant (and topological) quantum computing (2006-07) • quant-ph/0703264 (Aliferis, Gottesman, Preskill) Accuracy threshold for postselected quantum computation. Proof applies to a scheme where states prepared offline are protected by an error-detecting code. Established lower bound on the accuracy threshold, 1.04  10-3, the highest proved so far. • quant-ph/0610063 (Aliferis, Cross)Subsystem fault tolerance with the Bacon-Shor code. Codes with an unfixed gauge freedom lead to a highly efficient method for fault-tolerant error correction that can be implemented using only nearest-neighbor two-qubit measurements. • quant-ph/0703143 (Raussendorf, Harrington, Goyal) Topological fault-tolerance in cluster state quantum computation. Topologically protected quantum gates are realized by performing measurements that impose appropriate boundary conditions on a three-dimensional “cluster state.” (The spatial dimensionality can be reduced to two by converting one spatial axis of the cluster into time.) • quant-ph/0608119 (Bonderson, Shtengel, Slingerland)Decoherence of anyonic charge in interferometry measurements.Measuring anyonic charge using a Mach-Zehnder interferometer. Visibility of the interference is related to the topological S-matrix of the anyon model. • cond-mat/0611412 (Zwolak)Numerical ansatz for solving integro-differential equation with increasingly smooth memory kernels: spin-boson model and beyond. New method for studying real-time dynamics of systems that are strongly coupled to the environment; reduces computational cost of simulation.

6. Subsystem codes Hilbert space decomposes as: “gauge” subsystem syndrome “logical” subsystem • A subsystem code becomes a standard stabilizer code when the gauge subsystem is trivial (e.g., if we “fix the gauge”). • But there is no need to fix the gauge, as errors acting on gauge qubits do not damage the protected information. • Maintaining the gauge freedom reduces the number of check operators. • Syndrome information can be extracted by measuring the gauge qubits, and for some codes the gauge-qubit operators have lower weight than the stabilizer generators, so it is easier to measure the gauge operators fault tolerantly.

7. 3  3 “Bacon-Shor code” Shor (1995) Bacon (2005) check operators: logical operators: corrects one error ZZ ZZ These weight-two gauge Pauli operators commute with the logical operations, and measuring them determines the check operators in the stabilizer. Because only weight-two operators are measured, error correction is efficient and easily made fault tolerant. XX XX XX ZZ ZZ XX XX XX ZZ ZZ

8. 3  3 “Bacon-Shor code” Shor (1995) Bacon (2005) check operators: logical operators: corrects one error ZZ ZZ The optimal threshold estimate is found using the 5 X 5 Bacon-Shor code (which corrects two errors): XX XX XX ZZ ZZ XX XX XX ZZ ZZ Aliferis, Cross (2006)

9. Accuracy threshold using error-detecting codes error detect decode space time Using Bacon-Shor codes, we obtain a lower bound on the accuracy threshold (for adversarial local stochastic noise, nonlocal gates) 0 > 1.9  10-4 We can improve the threshold further if we can simulate gates with eff<0 using gates with  > 0. Knill’s idea (2004): Prepare suitable ancillas offline and teleport gates. Encoded error rate eff<0 can be achieved if the errors in the ancilla are nearly independent and have error rate below e.g. 5%. gate to be simulated encoded data in Bellmeas. entangledencoded ancilla encoded data out Protect the ancilla-preparation circuit using a (recursive) error-detecting code and accept the ancilla only if no errors are detected. Errors occurring during decoding are independent.

10. Threshold for postselected quantum computation We can boost the reliability by building a hierarchy of gadgets within gadgets --- the fault-tolerant circuit simulates the ideal circuit if the faults are sparse. However … to assess the reliability of the postselected circuit, we must estimate the probability that it fails conditioned on global acceptance --- i.e., acceptance by every error detection in the entire circuit. circuit fails here To obtain a threshold theorem for postselected computation, we must disallow correlations in the noise that could be tolerated if error correction were used instead. Otherwise, the devil could enhance greatly the conditional probability of failure in one part of the circuit by turning off faults elsewhere. Devil turns off faults elsewhere to enhance probability of failure conditioned on global acceptance.

11. Threshold for postselected quantum computation good bad bad bad bad bad The bad gadgets in the postselected circuit form connected clusters, surrounded by error detections with no faults. Thus the clusters (which typically contain just one or a small number of bad gadgets) are isolated from one another, enabling us to relate the probability of failure of a gadget conditioned on local acceptance (within the cluster) to its probability of failure conditioned on global acceptance. This means that error detection and (global) postselection improves reliability, and we can show by an inductive step that the probability of failure in a recursive simulation gets arbitrarily small if the noise is sufficiently weak.. Counting the ways for error-detecting gadgets to fail, we find 0,ED > 1.04  10-3 (Aliferis-Gottesman-Preskill 2007, Reichardt 2006). This is the best rigorously established lower bound on the accuracy threshold so far, but still a factor of 30 below Knill’s estimate based on simulations. (Overhead cost of the simulation can be improved by using the error-detecting codes to correct errors at known locations).

12. singularity message Bob radiation horizon Alice Are black holes quantum cloners? Reconciling the viewpoints of “inside” and “outside” observers is very subtle, but no observer should be able to detect a violation of the principles of quantum mechanics. Disturbingly, if Alice’s qubits are absorbed by the black hole and re-emitted in the Hawking radiation, these qubits seem to be in two places at once, a violation of the “no-cloning” principle. Can the cloning be verified? Suppose Bob recovers Alice’s qubits, which have become encoded in the Hawking radiation, after Schwarschild time t, and then jumps into the black hole to compare notes with Alice. If t = O(M log M) then Alice must send her message to Bob within proper time tPlanck after horizon crossing.

13. Bob decodes radiation black hole strongly mixing unitary radiation black hole maximal entanglement Alice’s qubits Hayden, Preskill How fast does information escape from a black hole? Once Alice’s k qubits are thoroughly mixed with the black hole’s internal degrees of freedom, the information escapes quickly --- as soon as k+constant qubits are emitted (assuming that evaporation is already past the half-way point, so that the black hole has become maximally entangled with the previously emitted radiation). Modeling the internal dynamics by a random local quantum circuit, we estimate the mixing time as t = O(M log M) Therefore Alice’s qubits escape on this time scale --- “black hole complementarity” just scrapes by!