54 research outputs found
Beyond the thermodynamic limit: finite-size corrections to state interconversion rates
Thermodynamics is traditionally constrained to the study of macroscopic
systems whose energy fluctuations are negligible compared to their average
energy. Here, we push beyond this thermodynamic limit by developing a
mathematical framework to rigorously address the problem of thermodynamic
transformations of finite-size systems. More formally, we analyse state
interconversion under thermal operations and between arbitrary
energy-incoherent states. We find precise relations between the optimal rate at
which interconversion can take place and the desired infidelity of the final
state when the system size is sufficiently large. These so-called second-order
asymptotics provide a bridge between the extreme cases of single-shot
thermodynamics and the asymptotic limit of infinitely large systems. We
illustrate the utility of our results with several examples. We first show how
thermodynamic cycles are affected by irreversibility due to finite-size
effects. We then provide a precise expression for the gap between the
distillable work and work of formation that opens away from the thermodynamic
limit. Finally, we explain how the performance of a heat engine gets affected
when one of the heat baths it operates between is finite. We find that while
perfect work cannot generally be extracted at Carnot efficiency, there are
conditions under which these finite-size effects vanish. In deriving our
results we also clarify relations between different notions of approximate
majorisation.Comment: 31 pages, 10 figures. Final version, to be published in Quantu
Tailoring surface codes for highly biased noise
The surface code, with a simple modification, exhibits ultra-high error
correction thresholds when the noise is biased towards dephasing. Here, we
identify features of the surface code responsible for these ultra-high
thresholds. We provide strong evidence that the threshold error rate of the
surface code tracks the hashing bound exactly for all biases, and show how to
exploit these features to achieve significant improvement in logical failure
rate. First, we consider the infinite bias limit, meaning pure dephasing. We
prove that the error threshold of the modified surface code for pure dephasing
noise is , i.e., that all qubits are fully dephased, and this threshold
can be achieved by a polynomial time decoding algorithm. We demonstrate that
the sub-threshold behavior of the code depends critically on the precise shape
and boundary conditions of the code. That is, for rectangular surface codes
with standard rough/smooth open boundaries, it is controlled by the parameter
, where and are dimensions of the surface code lattice. We
demonstrate a significant improvement in logical failure rate with pure
dephasing for co-prime codes that have , and closely-related rotated
codes, which have a modified boundary. The effect is dramatic: the same logical
failure rate achievable with a square surface code and physical qubits can
be obtained with a co-prime or rotated surface code using only
physical qubits. Finally, we use approximate maximum likelihood decoding to
demonstrate that this improvement persists for a general Pauli noise biased
towards dephasing. In particular, comparing with a square surface code, we
observe a significant improvement in logical failure rate against biased noise
using a rotated surface code with approximately half the number of physical
qubits.Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new
results on the performance of surface codes in the finite-bias regime,
obtained with beveled surface codes and an improved tensor network decoder;
v3 published versio
Quantum-embeddable stochastic matrices
The classical embeddability problem asks whether a given stochastic matrix
, describing transition probabilities of a -level system, can arise from
the underlying homogeneous continuous-time Markov process. Here, we investigate
the quantum version of this problem, asking of the existence of a Markovian
quantum channel generating state transitions described by a given . More
precisely, we aim at characterising the set of quantum-embeddable stochastic
matrices that arise from memoryless continuous-time quantum evolution. To this
end, we derive both upper and lower bounds on that set, providing new families
of stochastic matrices that are quantum-embeddable but not
classically-embeddable, as well as families of stochastic matrices that are not
quantum-embeddable. As a result, we demonstrate that a larger set of transition
matrices can be explained by memoryless models if the dynamics is allowed to be
quantum, but we also identify a non-zero measure set of random processes that
cannot be explained by either classical or quantum memoryless dynamics.
Finally, we fully characterise extreme stochastic matrices (with entries given
only by zeros and ones) that are quantum-embeddable.Comment: 14 pages, 3 figures, comments welcom
Tailoring three-dimensional topological codes for biased noise
Tailored topological stabilizer codes in two dimensions have been shown to
exhibit high storage threshold error rates and improved subthreshold
performance under biased Pauli noise. Three-dimensional (3D) topological codes
can allow for several advantages including a transversal implementation of
non-Clifford logical gates, single-shot decoding strategies, parallelized
decoding in the case of fracton codes as well as construction of fractal
lattice codes. Motivated by this, we tailor 3D topological codes for enhanced
storage performance under biased Pauli noise. We present Clifford deformations
of various 3D topological codes, such that they exhibit a threshold error rate
of under infinitely biased Pauli noise. Our examples include the 3D
surface code on the cubic lattice, the 3D surface code on a checkerboard
lattice that lends itself to a subsystem code with a single-shot decoder, the
3D color code, as well as fracton models such as the X-cube model, the
Sierpinski model and the Haah code. We use the belief propagation with ordered
statistics decoder (BP-OSD) to study threshold error rates at finite bias. We
also present a rotated layout for the 3D surface code, which uses roughly half
the number of physical qubits for the same code distance under appropriate
boundary conditions. Imposing coprime periodic dimensions on this rotated
layout leads to logical operators of weight at infinite bias and a
corresponding subthreshold scaling of the logical failure rate,
where is the number of physical qubits in the code. Even though this
scaling is unstable due to the existence of logical representations with
low-rate Pauli errors, the number of such representations scales only
polynomially for the Clifford-deformed code, leading to an enhanced effective
distance.Comment: 51 pages, 34 figure
Is the Welfare State Sustainable? Experimental Evidence on Citizens' Preferences for Redistribution
The sustainability of the welfare state ultimately depends on citizens' preferences for income redistribution. They are elicited through a Discrete Choice Experiment performed in 2008 in Switzerland. Attributes are redistribution as GDP share, its uses (the unemployed, old-age pensioners, people with ill health etc.), and nationality of beneficiary. Estimated marginal willingness to pay (WTP) is positive among those who deem benefits too low, and negative otherwise. However, even those who state that government should reduce income inequality exhibit a negative WTP on average. The major finding is that estimated average WTP is maximum at 21% of GDP, clearly below the current value of 25%. Thus, the present Swiss welfare state does not appear sustainable
Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics
We address the problem of exact and approximate transformation of quantum
dichotomies in the asymptotic regime, i.e., the existence of a quantum channel
mapping into with an
error (measured by trace distance) and into
exactly, for a large number . We derive
second-order asymptotic expressions for the optimal transformation rate
in the small, moderate, and large deviation error regimes, as well as the
zero-error regime, for an arbitrary pair of initial states
and a commuting pair of final states. We also prove that
for and given by thermal Gibbs states, the derived
optimal transformation rates in the first three regimes can be attained by
thermal operations. This allows us, for the first time, to study the
second-order asymptotics of thermodynamic state interconversion with fully
general initial states that may have coherence between different energy
eigenspaces. Thus, we discuss the optimal performance of thermodynamic
protocols with coherent inputs and describe three novel resonance phenomena
allowing one to significantly reduce transformation errors induced by
finite-size effects. What is more, our result on quantum dichotomies can also
be used to obtain, up to second-order asymptotic terms, optimal conversion
rates between pure bipartite entangled states under local operations and
classical communication.Comment: 51 pages, 6 figures, comments welcom
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