Format results
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Resource dependence relations
Yìlè Yīng Perimeter Institute for Theoretical Physics
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Quantum metrology with correlated noise
Stanisław Kurdziałek University of Warsaw
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Holographic quantum tasks in the static patch
Victor Franken École Polytechnique
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Symmetry enforced entanglement in maximally mixed states
Subhayan Sahu Perimeter Institute for Theoretical Physics
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Topological Quantum Information, Khovanov Homology and the Jones Polynomial
Louis Kauffman University of Illinois at Chicago
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Constant-Overhead Magic State Distillation
Hayata YamasakiMagic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol’s overhead, defined as the number of input magic states required per output magic state with an error rate below ϵ, typically grows as O(log^γ (1/ϵ)) as ϵ → 0. Achieving smaller overheads, i.e., smaller exponents γ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ > 0, and identifying the smallest achievable exponent γ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1) overhead, meaning the optimal exponent γ = 0; this improves over the previous best of γ ≈ 0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 2^10-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. These codes can be realized by representing each 2^10-dimensional qudit as a set of 10 qubits, using stabilizer operations on qubits. We prove that the use of asymptotically good codes with non-vanishing rate and relative distance in magic state distillation leads to the constant overhead. The 10-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-Z (CCZ) and T gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent. The talk is based on the following paper. https://arxiv.org/abs/2408.07764 -
Generalized Quantum Stein's Lemma and Second Law of Quantum Resource Theories
Hayata YamasakiThe second law of thermodynamics is the cornerstone of physics, characterizing the convertibility between thermodynamic states through a single function, entropy. Given the universal applicability of thermodynamics, a fundamental question in quantum information theory is whether an analogous second law can be formulated to characterize the convertibility of resources for quantum information processing by a single function. In 2008, a promising formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing. Central to this formulation was the generalized quantum Stein's lemma, which aimed to characterize this optimal performance by a measure of quantum resources, the regularized relative entropy of resource. If proven valid, the generalized quantum Stein's lemma would lead to the second law for quantum resources, with the regularized relative entropy of resource taking the role of entropy in thermodynamics. However, in 2023, a logical gap was found in the original proof of this lemma, casting doubt on the possibility of such a formulation of the second law. In this work, we address this problem by developing alternative techniques to successfully prove the generalized quantum Stein's lemma under a smaller set of assumptions than the original analysis. Based on our proof, we reestablish and extend the second law of quantum resource theories, applicable to both static resources of quantum states and a fundamental class of dynamical resources represented by classical-quantum (CQ) channels. These results resolve the fundamental problem of bridging the analogy between thermodynamics and quantum information theory. The talk is based on the following paper. https://arxiv.org/abs/2408.02722 -
Random unitaries in extremely low depth
Thomas SchusterRandom unitaries form the backbone of numerous components of quantum technologies, and serve as indispensable toy models for complex processes in quantum many-body physics. In all of these applications, a crucial consideration is in what circuit depth a random unitary can be generated. I will present recent work, in which we show that local quantum circuits can form random unitaries in exponentially lower circuit depths than previously thought. We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over n qubits in log n depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in poly log n depth, and in all-to-all-connected circuits in poly log log n depth. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.
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Resource dependence relations
Yìlè Yīng Perimeter Institute for Theoretical Physics
A resource theory imposes a preorder over states, with one state being above another if the first can be converted to the second by a free operation, and where the set of free operations defines the notion of resourcefulness under study. In general, the location of a state in the preorder of one resource theory can constrain its location in the preorder of a different resource theory. It follows that there can be nontrivial dependence relations between different notions of resourcefulness. In this talk, we lay out the conceptual and formal groundwork for the study of resource dependence relations. In particular, we note that the relations holding among a set of monotones that includes a complete set for each resource theory provides a full characterization of resource dependence relations. As an example, we consider three resource theories concerning the about-face asymmetry properties of a qubit along three mutually orthogonal axes on the Bloch ball, where about-face symmetry refers to a representation of $\mathbb{Z}_2$, consisting of the identity map and a $\pi$ rotation about the given axis. This example is sufficiently simple that we are able to derive a complete set of monotones for each resource theory and to determine all of the relations that hold among these monotones, thereby completely solving the problem of determining resource dependence relations. Nonetheless, we show that even in this simplest of examples, these relations are already quite nuanced. At the end of the talk, we will briefly discuss how to witness nonclassicality in quantum resource dependence relations and demonstrate it with the about-face asymmetry example. The talk is based on the preprint: arXiv:2407.00164 and ongoing work. -
Quantum metrology with correlated noise
Stanisław Kurdziałek University of Warsaw
I will present a universal numerical tool for identifying optimal adaptive metrological protocols in the presence of both uncorrelated and correlated noise [arXiv:2403.04854]. Leveraging a novel tensor network decomposition of quantum combs, the algorithm demonstrates efficiency even with a large number of channel uses (N=50). In the second part of the talk, I will explore the generalization of existing metrological upper bounds [Nat. Com. 3, 1063 (2012), PRL 131(9), 090801 (2023)] for correlated noise scenarios [arXiv:2410.01881].
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Holographic quantum tasks in the static patch
Victor Franken École Polytechnique
Static patch holography is a conjectured duality between the static patch of an observer in de Sitter spacetime and a quantum theory defined on its (stretched) cosmological horizon. We illustrate from entanglement wedge reconstruction how a closed and connected de Sitter spacetime can emerge in this framework from the entanglement between the two holographic screens of two antipodal observers. In holographic spacetimes, a direct scattering in the bulk may not have a local boundary analog, imposing the existence of O(1/G) mutual information on the boundary. This statement is formalized by the connected wedge theorem, which is expected to hold beyond the AdS/CFT correspondence from which it originates. We consider scatterings in the static patch of an observer. We argue that for static patch holography to be consistent with the connected wedge theorem, causality on the stretched horizon should be induced from null infinity. In particular, signals propagating in the static patch are associated with fictitious local operators at null infinity. We present a sketch of proof of the connected wedge theorem in asymptotically de Sitter spacetime using induced causality.
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Symmetry enforced entanglement in maximally mixed states
Subhayan Sahu Perimeter Institute for Theoretical Physics
Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. SU(2)), the bipartite entanglement of formation for the MMIS scales logarithmically ∼logN with the number of qudits N.
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Topological Quantum Information, Khovanov Homology and the Jones Polynomial
Louis Kauffman University of Illinois at Chicago
In this talk (based on arXiv:1001.0354) we give a quantum statistical interpretation for the Kauffmann bracket polynomial state sum <K> for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.