Quantum Error Correction and Fault-Tolerance Todd A. Brun, Daniel A. Lidar, Ben Reichardt, Paolo Zanardi University of Southern California

The key to quantum computation The most serious obstacle to realizing quantum computers is decoherence. For almost all plausible physical systems, the same coupling to the outside world that allows controlled unitary evolution opens the system up to noise. Quantum information must therefore be protected by quantum error correction; and errors must be contained by fault-tolerant design.

The most widely studied approaches to fault- tolerant quantum computing (FTQC) use either concatenated quantum error correcting codes (QECCs) or topological codes (TCs), such as the surface code. Here we look briefly at some additional approaches that we believe are highly promising, and may overcome some of the shortcomings of other methods.

Multi-qubit Block Codes 1.When not being processed, logical qubits are stored in [[n,k,d]] block codes. These codes are corrected by repeatedly measuring the stabilizer generators via Steane extraction. 2.Logical Clifford gates can be done by measuring sequences of logical operators. This is also done by Steane extraction with a modified ancilla state. 3.A non-Clifford gate is done by teleporting the selected logical qubits into the processor block, and teleporting back after the gate. Teleportation is also used between storage blocks. 4.The processor blocks use codes that allow transversal circuits for the encoded gates. E.g., for the T gate we could use the concatenated [[15,1,3]] shortened Reed-Muller code. 5.Logical teleportation is also done by measuring a sequence of logical operators. Brun, Zheng, Hsu, Job and Lai, in preparation.

Outline of the scheme

Codes with universal transversal sets Normally codes do not allow a complete universal set of gates to be performed transversally. Universality requires additional resources (e.g., magic state distillation). However, one can find pairs of operator codes, one of which allows transversal Hadamard gates and one T gates, in which one can transform back and forth between them by measuring syndrome generators. (E.g., 3D color codes.) Paetznick and Reichardt

Fault-tolerant Holonomic QC In Holonomic QC, the computer is always in the ground state of a Hamiltonian. Quantum gates are done by adiabatically deforming the Hamiltonian, exploiting holonomies of the ground space family. These schemes can be made fault-tolerant by combining with a QECC. Extensions can be made to nonadiabatic evolutions, and solutions that combine topological and holonomic robustness.

E.g., We can do HQC in surface codes.

Stabilization and error suppression It is also possible to suppress the effects of errors under certain circumstances. For example, noise processes with intrinsic symmetries may allow decoherence-free subspaces or noiseless subsystems. In some cases, the effects of noise can be undone or canceled, for instance by dynamical decoupling.

Dissipation-Assisted Quantum Information Processing Paolo Zanardi and Daniel A. Lidar

Dissipative dynamics: quantum coherence/entanglement Are (typically & quickly) Destroyed = Worst enemy of QIP Fight with: quantum error correction, decoherence-free (DFS) Subspaces, dynamical-decoupling, topological-geometric QIP,… “ Traditional View ” “ Novel View ” Dissipative dynamics can be harnessed & exploited to the end of QIP: pure-state preparation, quantum computation, quantum simulations Our Strategy is based on Dissipation-Assisted Quantum Information Processing Paolo Zanardi & Daniel Lidar (USC) Dissipation-Assisted Quantum Information Processing Paolo Zanardi & Daniel Lidar (USC) P. Zanardi, L. Campos Venuti, Phys. Rev. Lett. 113, (2014)

Coherent holonomic (geometric) manipulations in the SSM Complexity, (non) locality and robustness can be enhanced Moral: dissipation & decoherence can be good guys… We have shown dissipation-assisted gates in DFSs and NSs Projection Theorem: small Hamiltonian controls gets Zeno-projected and Renormalized by strong dissipation and give rise to an effective unitary dynamics over the Steady State Manifold (SSM) of the system Projection Theorem: small Hamiltonian controls gets Zeno-projected and Renormalized by strong dissipation and give rise to an effective unitary dynamics over the Steady State Manifold (SSM) of the system …and DOE should not forget about that! Thanks Todd! Unitary dynamics can emerge out of purely dissipative one

Error Correction in Quantum Annealing and Adiabatic QC

Quantum Annealing Correction Strategy 1. Encode into repetition code: 2. Add energy penalty: 3. Combine: 4. Run QA: 5. decode at end by “majority vote” “penalty scale” (optimized)

Quantum Annealing Correction works penalty + majority voting on encoded qubits K. Pudenz, T. Albash, D.A. Lidar, Nature Comm. 5, 3243 (2014) α=0.3 best of 4 chains run in parallel

Also random Ising problems benefit [arXiv: ] The advantage gained from QAC is larger for larger and harder problems number of repetitions to find solution at least once α=0.5

Thank you for your attention!

Estimated Performance of Storage Blocks [[2047,23,63]] (blue), [[2921,57,77]] (red), and [[5865,143,105]] (green). These curves are generated by Monte Carlo simulations and linear extrapolation. However, the extrapolation at low error rates can be backed up by an upper bound calculated purely from the distances of the code. For p eff = 0.007, we get bounds on the error rates of approximately , 2.5x10 -19, and 7x

The [[15,1,3]] code at two (blue) and three (red) levels of concatenation. A combination of Monte Carlo simulations at higher errors, a rough bound at lower errors, and (not to be relied upon) extrapolation. At p eff = (all contributing error processes below 5x10 -4 ), the block error rate is estimated to be roughly 2x This is certainly small enough to carry out highly nontrivial quantum computations. Estimated Performance of Processor Blocks