All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun, Joseph Ben; Ramgoolam, Sanjaye

doi:10.1088/1751-8121/ac9b3b

Cited by 5 publications

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References 47 publications

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“…The dimensions of these Hilbert spaces grow as n! at large n (see [65] for all orders asymptotic formulae for these dimensions). An important motivation for this paper has been to take advantage of known exponential improvements due to quantum algorithms [58] to formulae efficient algorithms based on K(n) for Kronecker coefficients, since an important element of the interest in Kronecker coefficients comes from computational complexity theory [66][67][68][69].…”

Section: More General Choices Of Vmentioning

confidence: 98%

The quantum detection of projectors in finite-dimensional algebras and holography

Geloun

Ramgoolam

2023

J. High Energ. Phys.

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We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes n. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of n. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having n boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in n. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with m, n and m + n boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in m and n.

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Section: More General Choices Of Vmentioning

confidence: 98%

The quantum detection of projectors in finite-dimensional algebras and holography

Geloun

Ramgoolam

2023

J. High Energ. Phys.

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Gauged permutation invariant matrix quantum mechanics: partition functions

O’Connor,

Ramgoolam

2024

J. High Energ. Phys.

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The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

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Counting $$U(N)^{\otimes r}\otimes O(N)^{\otimes q}$$ invariants and tensor model observables

Avohou,

Ben Geloun,

Toriumi

2024

Eur. Phys. J. C

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Abstract$$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ U ( N ) ⊗ r ⊗ O ( N ) ⊗ q invariants are constructed by contractions of complex tensors of order $$r+q$$ r + q , also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ U ( N ) ⊗ r ⊗ O ( N ) ⊗ q invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $$r>1$$ r > 1 , the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At $$r=1$$ r = 1 , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ q + 3 branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.

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All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Cited by 5 publications

References 47 publications

The quantum detection of projectors in finite-dimensional algebras and holography

The quantum detection of projectors in finite-dimensional algebras and holography

Gauged permutation invariant matrix quantum mechanics: partition functions

Counting $$U(N)^{\otimes r}\otimes O(N)^{\otimes q}$$ invariants and tensor model observables

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