15 research outputs found
Overhead analysis of universal concatenated quantum codes
We analyze the resource overhead of recently proposed methods for universal fault-tolerant quantum computation using concatenated codes. Namely, we examine the concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, which allows for the construction of the 49- and 105-qubit codes that do not require the need for magic state distillation for universality. We compute a lower bound for the adversarial noise threshold of the 105-qubit code and find it to be 8.33 × 10(−6). We obtain a depolarizing noise threshold for the 49-qubit code of 9.69 × 10(−4) which is competitive with the 105-qubit threshold result of 1.28 × 10^(−3). We then provide lower bounds on the resource requirements of the 49- and 105-qubit codes and compare them with the surface code implementation of a logical T gate using magic state distillation. For the sampled input error rates and noise model, we find that the surface code achieves a smaller overhead compared to our concatenated schemes
Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces
The \emph{Chow parameters} of a Boolean function
are its degree-0 and degree-1 Fourier coefficients. It has been known
since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of
any linear threshold function uniquely specify within the space of all
Boolean functions, but until recently (O'Donnell and Servedio) nothing was
known about efficient algorithms for \emph{reconstructing} (exactly or
approximately) from exact or approximate values of its Chow parameters. We
refer to this reconstruction problem as the \emph{Chow Parameters Problem.}
Our main result is a new algorithm for the Chow Parameters Problem which,
given (sufficiently accurate approximations to) the Chow parameters of any
linear threshold function , runs in time \tilde{O}(n^2)\cdot
(1/\eps)^{O(\log^2(1/\eps))} and with high probability outputs a
representation of an LTF that is \eps-close to . The only previous
algorithm (O'Donnell and Servedio) had running time \poly(n) \cdot
2^{2^{\tilde{O}(1/\eps^2)}}.
As a byproduct of our approach, we show that for any linear threshold
function over , there is a linear threshold function which
is \eps-close to and has all weights that are integers at most \sqrt{n}
\cdot (1/\eps)^{O(\log^2(1/\eps))}. This significantly improves the best
previous result of Diakonikolas and Servedio which gave a \poly(n) \cdot
2^{\tilde{O}(1/\eps^{2/3})} weight bound, and is close to the known lower
bound of (1/\eps)^{\Omega(\log \log (1/\eps))}\} (Goldberg,
Servedio). Our techniques also yield improved algorithms for related problems
in learning theory
Fault-tolerant magic state preparation with flag qubits
Magic state distillation is one of the leading candidates for implementing
universal fault-tolerant logical gates. However, the distillation circuits
themselves are not fault-tolerant, so there is additional cost to first
implement encoded Clifford gates with negligible error. In this paper we
present a scheme to fault-tolerantly and directly prepare magic states using
flag qubits. One of these schemes uses a single extra ancilla, even with noisy
Clifford gates. We compare the physical qubit and gate cost of this scheme to
the magic state distillation protocol of Meier, Eastin, and Knill, which is
efficient and uses a small stabilizer circuit. In some regimes, we show that
the overhead can be improved by several orders of magnitude.Comment: 26 pages, 17 figures, 5 tables. Comments welcome! v2 (published
version): quantumarticle documentclass and expanded discussions on the
fault-tolerant scheme
Numerical and analytical studies of quantum error correction
A reliable large-scale quantum computer, if built, can solve many real-life problems exponentially faster than the existing digital devices. The biggest obstacle to building one is that they are extremely sensitive and error-prone regardless of the selection of physical implementation. Both data storage and data manipulation require careful implementation and precise control due to its quantum mechanical nature. For the development of a practical and scalable computer, it is essential to identify possible quantum errors and reduce them throughout every layer of the hierarchy of quantum computation.
In this dissertation, we present our investigation into new methods to reduce errors in quantum computers from three different directions: quantum memory, quantum control, and quantum error correcting codes. For quantum memory, we pursue the potential of the quantum equivalent of a magnetic hard drive using two-body-interaction structures in fractal dimensions. With regard to quantum control, we show that it is possible to arbitrarily reduce error when manipulating multiple quantum bits using a technique popular in nuclear magnetic resonance. Finally, we introduce an efficient tool to study quantum error correcting codes and present analysis of the codes' performance on model quantum architectures.Ph.D
Quantum computing is scalable on a planar array of qubits with fabrication defects
To successfully execute large-scale algorithms, a quantum computer will need to perform its elementary operations near perfectly. This is a fundamental challenge since all physical qubits suffer a considerable level of noise. Moreover, real systems are likely to have a finite yield, i.e., some nonzero proportion of the components in a complex device may be irredeemably broken at the fabrication stage. We present a threshold theorem showing that an arbitrarily large quantum computation can be completed with a vanishing probability of failure using a two-dimensional array of noisy qubits with a finite density of fabrication defects. To complete our proof we introduce a robust protocol to measure high-weight stabilizers to compensate for large regions of inactive qubits. We obtain our result using a surface-code architecture. Our approach is therefore readily compatible with ongoing experimental efforts to build a large-scale quantum computer
Quantum computing is scalable on a planar array of qubits with fabrication defects
To successfully execute large-scale algorithms, a quantum computer will need to perform its elementary operations near perfectly. This is a fundamental challenge since all physical qubits suffer a considerable level of noise. Moreover, real systems are likely to have a finite yield, i.e., some nonzero proportion of the components in a complex device may be irredeemably broken at the fabrication stage. We present a threshold theorem showing that an arbitrarily large quantum computation can be completed with a vanishing probability of failure using a two-dimensional array of noisy qubits with a finite density of fabrication defects. To complete our proof we introduce a robust protocol to measure high-weight stabilizers to compensate for large regions of inactive qubits. We obtain our result using a surface-code architecture. Our approach is therefore readily compatible with ongoing experimental efforts to build a large-scale quantum computer