We accept the implicit challenge of A. Uhlmann in his 1994 paper, "Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits," by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, "Geometric Phases for Mixed States in Interferometry" (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjöqvist et al method. Additional phenomena of interest are reported too.We accept the implicit challenge of A. Uhlmann in his 1994 paper, "Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits," by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, "Geometric Phases for Mixed States in Interferometry" (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjöqvist et al method. Additional phenomena of interest are reported too.

arXiv:math-ph/0112054Dec 21, 20012 citations

Geometric Phases of the Uhlmann Type for n-Level Gibbsian Density Matrices

Paul B. Slater

We accept the implicit challenge of A. Uhlmann in his 1994 paper, Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits, by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, Geometric Phases for Mixed States in Interferometry (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjoqvist et al method. Additional phenomena of interest are reported too.We accept the implicit challenge of A. Uhlmann in his 1994 paper, Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits, by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, Geometric Phases for Mixed States in Interferometry (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjoqvist et al method. Additional phenomena of interest are reported too.

Dec 21, 20010 citations

Lifted surgery: Fast processing with QLDPC codes

Lucas Berent, Lawrence Z. Cohen, Armanda O. Quintavalle

Quantum low-density parity-check (QLDPC) codes are a leading candidate for achieving low-overhead fault-tolerant quantum computing. However, the time overhead of logical operations in QLDPC codes remains a key challenge. Code surgery, a space-efficient technique for fault-tolerant logical measurements, incurs this overhead through repeated rounds of syndrome measurement. We introduce lifted surgery, a method for fast and parallel surgery on Abelian group algebra codes that maintains the low physical overhead that makes QLDPC codes attractive. Lifted surgery preserves the symmetries of the underlying code, making searches for large instances tractable and offering a natural route towards efficient hardware implementations. We utilise block decompositions and techniques from commutative algebra to characterise lifted surgery, investigate well-behaved subfamilies, and construct explicit examples. In particular, we present quantum radial codes with parameters $[[90, 8, 10]]$ and $[[198, 8, 16 ]]$ for which surgery is fast, parallel, and addressable, allowing arbitrary sets of independent logical operators of the same Pauli type to be measured in a single round of syndrome extraction. We benchmark lifted surgery under circuit-level depolarising noise and, for the $[[ 90, 8, 10 ]]$ code, find logical performance comparable to standard code surgery while requiring ten times fewer rounds of syndrome measurement. By combining speed and parallelism, lifted surgery offers a practical route towards low-overhead fault-tolerant quantum computing.Quantum low-density parity-check (QLDPC) codes are a leading candidate for achieving low-overhead fault-tolerant quantum computing. However, the time overhead of logical operations in QLDPC codes remains a key challenge. Code surgery, a space-efficient technique for fault-tolerant logical measurements, incurs this overhead through repeated rounds of syndrome measurement. We introduce lifted surgery, a method for fast and parallel surgery on Abelian group algebra codes that maintains the low physical overhead that makes QLDPC codes attractive. Lifted surgery preserves the symmetries of the underlying code, making searches for large instances tractable and offering a natural route towards efficient hardware implementations. We utilise block decompositions and techniques from commutative algebra to characterise lifted surgery, investigate well-behaved subfamilies, and construct explicit examples. In particular, we present quantum radial codes with parameters $[[90, 8, 10]]$ and $[[198, 8, 16 ]]$ for which surgery is fast, parallel, and addressable, allowing arbitrary sets of independent logical operators of the same Pauli type to be measured in a single round of syndrome extraction. We benchmark lifted surgery under circuit-level depolarising noise and, for the $[[ 90, 8, 10 ]]$ code, find logical performance comparable to standard code surgery while requiring ten times fewer rounds of syndrome measurement. By combining speed and parallelism, lifted surgery offers a practical route towards low-overhead fault-tolerant quantum computing.

quant-pharXiv:2609.11723Sep 11, 20260 citations

[[90, 8, 10]]

and

[[198, 8, 16 ]]

for which surgery is fast, parallel, and addressable, allowing arbitrary sets of independent logical operators of the same Pauli type to be measured in a single round of syndrome extraction. We benchmark lifted surgery under circuit-level depolarising noise and, for the

[[ 90, 8, 10 ]]

code, find logical performance comparable to standard code surgery while requiring ten times fewer rounds of syndrome measurement. By combining speed and parallelism, lifted surgery offers a practical route towards low-overhead fault-tolerant quantum computing.