



01860nas a2200145   4500008004100000245007100041210006900112260001500181520139900196100001601595700002101611700002401632700002101656856003701677       2023                        eng d00aFault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes0 aFaultTolerant Quantum Memory using LowDepth Random Circuit Codes  c11/29/20233 a<p>Low-depth random circuit codes possess many desirable properties for quantum error correction but have so far only been analyzed in the code capacity setting where it is assumed that encoding gates and syndrome measurements are noiseless. In this work, we design a fault-tolerant distillation protocol for preparing encoded states of one-dimensional random circuit codes even when all gates and measurements are subject to noise. This is sufficient for fault-tolerant quantum memory since these encoded states can then be used as ancillas for Steane error correction. We show through numerical simulations that our protocol can correct erasure errors up to an error rate of 2%. In addition, we also extend results in the code capacity setting by developing a maximum likelihood decoder for depolarizing noise similar to work by Darmawan et al. As in their work, we formulate the decoding problem as a tensor network contraction and show how to contract the network efficiently by exploiting the low-depth structure. Replacing the tensor network with a so-called &#39;&#39;tropical&#39;&#39; tensor network, we also show how to perform minimum weight decoding. With these decoders, we are able to numerically estimate the depolarizing error threshold of finite-rate random circuit codes and show that this threshold closely matches the hashing bound even when the decoding is sub-optimal.</p>1 aNelson, Jon1 aBentsen, Gregory1 aFlammia, Steven, T.1 aGullans, Michael  uhttps://arxiv.org/abs/2311.1798502376nas a2200157   4500008004100000245005000041210004900091260001500140520192200155100002102077700002202098700002002120700001702140700002402157856003702181       2020                        eng d00aQuantum coding with low-depth random circuits0 aQuantum coding with lowdepth random circuits  c10/19/20203 a<p>Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity in D&ge;1 spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth O(logN) random circuit is necessary and sufficient to converge (with high probability) to zero failure probability for any finite amount below the channel capacity for any D. Previous results on random circuits have only shown that O(N1/D) depth suffices or that O(log3N) depth suffices for all-to-all connectivity (D&rarr;&infin;). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the channel capacity converge to zero with N. We find that the requisite depth scales like O(logN) only for dimensions D&ge;2, and that random circuits require O(N&minus;&minus;&radic;) depth for D=1. Finally, we introduce an &quot;expurgation&quot; algorithm that uses quantum measurements to remove logical operators that cause the code to fail by turning them into either additional stabilizers or into gauge operators in a subsystem code. With such targeted measurements, we can achieve sub-logarithmic depth in D&ge;2 spatial dimensions below capacity without increasing the maximum weight of the check operators. We find that for any rate beneath the capacity, high-performing codes with thousands of logical qubits are achievable with depth 4-8 expurgated random circuits in D=2 dimensions. These results indicate that finite-rate quantum codes are practically relevant for near-term devices and may significantly reduce the resource requirements to achieve fault tolerance for near-term applications.&nbsp;</p>1 aGullans, Michael1 aKrastanov, Stefan1 aHuse, David, A.1 aJiang, Liang1 aFlammia, Steven, T.  uhttps://arxiv.org/abs/2010.0977502431nas a2200169   4500008004100000245010800041210006900149260001500218300001100233490000700244520189900251100002402150700001702174700001602191700001702207856003702224       2012                        eng d00aQuantum Tomography via Compressed Sensing: Error Bounds, Sample  Complexity, and Efficient Estimators
0 aQuantum Tomography via Compressed Sensing Error Bounds Sample Co  c2012/09/27  a0950220 v143 a  Intuitively, if a density operator has small rank, then it should be easier
to estimate from experimental data, since in this case only a few eigenvectors
need to be learned. We prove two complementary results that confirm this
intuition. First, we show that a low-rank density matrix can be estimated using
fewer copies of the state, i.e., the sample complexity of tomography decreases
with the rank. Second, we show that unknown low-rank states can be
reconstructed from an incomplete set of measurements, using techniques from
compressed sensing and matrix completion. These techniques use simple Pauli
measurements, and their output can be certified without making any assumptions
about the unknown state.
  We give a new theoretical analysis of compressed tomography, based on the
restricted isometry property (RIP) for low-rank matrices. Using these tools, we
obtain near-optimal error bounds, for the realistic situation where the data
contains noise due to finite statistics, and the density matrix is full-rank
with decaying eigenvalues. We also obtain upper-bounds on the sample complexity
of compressed tomography, and almost-matching lower bounds on the sample
complexity of any procedure using adaptive sequences of Pauli measurements.
  Using numerical simulations, we compare the performance of two compressed
sensing estimators with standard maximum-likelihood estimation (MLE). We find
that, given comparable experimental resources, the compressed sensing
estimators consistently produce higher-fidelity state reconstructions than MLE.
In addition, the use of an incomplete set of measurements leads to faster
classical processing with no loss of accuracy.
  Finally, we show how to certify the accuracy of a low rank estimate using
direct fidelity estimation and we describe a method for compressed quantum
process tomography that works for processes with small Kraus rank.
1 aFlammia, Steven, T.1 aGross, David1 aLiu, Yi-Kai1 aEisert, Jens  uhttp://arxiv.org/abs/1205.2300v200981nas a2200133   4500008004100000245005900041210005900100260001300159490000800172520059000180100002400770700001600794856003700810       2011                        eng d00aDirect Fidelity Estimation from Few Pauli Measurements0 aDirect Fidelity Estimation from Few Pauli Measurements  c2011/6/80 v1063 a  We describe a simple method for certifying that an experimental device
prepares a desired quantum state rho. Our method is applicable to any pure
state rho, and it provides an estimate of the fidelity between rho and the
actual (arbitrary) state in the lab, up to a constant additive error. The
method requires measuring only a constant number of Pauli expectation values,
selected at random according to an importance-weighting rule. Our method is
faster than full tomography by a factor of d, the dimension of the state space,
and extends easily and naturally to quantum channels.
1 aFlammia, Steven, T.1 aLiu, Yi-Kai  uhttp://arxiv.org/abs/1104.4695v301775nas a2200229   4500008004100000245003900041210003900080260001500119300000800134490000600142520116900148100001901317700002301336700002401359700001701383700002601400700001901426700002901445700001601474700001801490856003701508       2010                        eng d00aEfficient quantum state tomography0 aEfficient quantum state tomography  c2010/12/21  a1490 v13 a  Quantum state tomography, the ability to deduce the state of a quantum system
from measured data, is the gold standard for verification and benchmarking of
quantum devices. It has been realized in systems with few components, but for
larger systems it becomes infeasible because the number of quantum measurements
and the amount of computation required to process them grows exponentially in
the system size. Here we show that we can do exponentially better than direct
state tomography for a wide range of quantum states, in particular those that
are well approximated by a matrix product state ansatz. We present two schemes
for tomography in 1-D quantum systems and touch on generalizations. One scheme
requires unitary operations on a constant number of subsystems, while the other
requires only local measurements together with more elaborate post-processing.
Both schemes rely only on a linear number of experimental operations and
classical postprocessing that is polynomial in the system size. A further
strength of the methods is that the accuracy of the reconstructed states can be
rigorously certified without any a priori assumptions.
1 aCramer, Marcus1 aPlenio, Martin, B.1 aFlammia, Steven, T.1 aGross, David1 aBartlett, Stephen, D.1 aSomma, Rolando1 aLandon-Cardinal, Olivier1 aLiu, Yi-Kai1 aPoulin, David  uhttp://arxiv.org/abs/1101.4366v101338nas a2200169   4500008004100000245005200041210005200093260001400145490000800159520087000167100001701037700001601054700002401070700002001094700001701114856003701131       2010                        eng d00aQuantum state tomography via compressed sensing0 aQuantum state tomography via compressed sensing  c2010/10/40 v1053 a  We establish methods for quantum state tomography based on compressed
sensing. These methods are specialized for quantum states that are fairly pure,
and they offer a significant performance improvement on large quantum systems.
In particular, they are able to reconstruct an unknown density matrix of
dimension d and rank r using O(rd log^2 d) measurement settings, compared to
standard methods that require d^2 settings. Our methods have several features
that make them amenable to experimental implementation: they require only
simple Pauli measurements, use fast convex optimization, are stable against
noise, and can be applied to states that are only approximately low-rank. The
acquired data can be used to certify that the state is indeed close to pure, so
no a priori assumptions are needed. We present both theoretical bounds and
numerical simulations.
1 aGross, David1 aLiu, Yi-Kai1 aFlammia, Steven, T.1 aBecker, Stephen1 aEisert, Jens  uhttp://arxiv.org/abs/0909.3304v4