Permutation-invariant qudit codes from polynomials

A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also correct $t$ errors for $d \ge 2$. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of $q-1$ real polynomials that satisfy some combinatorial constraints. When $N > (2t+1)^2(d-1)$, we prove constructively that an uncountable number of such codes exist.Comment: 14 pages. Minor corrections made, to appear in Linear Algebra and its Application

Permutation-invariant codes encoding more than one qubit

A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation invariant codes and quantum error correction.Comment: 4 pages, minor change

Robust projective measurements through measuring code-inspired observables

Quantum measurements are ubiquitous in quantum information processing tasks, but errors can render their outputs unreliable. Here, we present a scheme that implements a robust projective measurement through measuring code-inspired observables. Namely, given a projective POVM, a classical code and a constraint on the number of measurement outcomes each observable can have, we construct commuting observables whose measurement is equivalent to the projective measurement in the noiseless setting. Moreover, we can correct $t$ errors on the classical outcomes of the observables' measurement if the classical code corrects $t$ errors. Since our scheme does not require the encoding of quantum data onto a quantum error correction code, it can help construct robust measurements for near-term quantum algorithms that do not use quantum error correction. Moreover, our scheme works for any projective POVM, and hence can allow robust syndrome extraction procedures in non-stabilizer quantum error correction codes.Comment: 7 pages, 1 figure, 2 column

Tight Cram\'{e}r-Rao type bounds for multiparameter quantum metrology through conic programming

In the quest to unlock the maximum potential of quantum sensors, it is of paramount importance to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. However, it is still not known how to find practical measurements with optimal precisions, even for uncorrelated measurements over probe states. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions. We solve this fundamental problem by introducing a framework of conic programming that unifies the theory of precision bounds for multiparameter estimates for uncorrelated and correlated measurement strategies under a common umbrella. Namely, we give precision bounds that arise from linear programs on various cones defined on a tensor product space of matrices, including a particular cone of separable matrices. Subsequently, our theory allows us to develop an efficient algorithm that calculates both upper and lower bounds for the ultimate precision bound for uncorrelated measurement strategies, where these bounds can be tight. In particular, the uncorrelated measurement strategy that arises from our theory saturates the upper bound to the ultimate precision bound. Also, we show numerically that there is a strict gap between the previous efficiently computable bounds and the ultimate precision bound.Comment: 23 pages, 5 figure

Weight Distribution of Classical Codes Influences Robust Quantum Metrology

Quantum metrology (QM) is expected to be a prominent use-case of quantum technologies. However, noise easily degrades these quantum probe states, and negates the quantum advantage they would have offered in a noiseless setting. Although quantum error correction (QEC) can help tackle noise, fault-tolerant methods are too resource intensive for near-term use. Hence, a strategy for (near-term) robust QM that is easily adaptable to future QEC-based QM is desirable. Here, we propose such an architecture by studying the performance of quantum probe states that are constructed from $[n,k,d]$ binary block codes of minimum distance $d \geq t+1$. Such states can be interpreted as a logical state of a CSS code whose logical $X$ group is defined by the aforesaid binary code. When a constant, $t$, number of qubits of the quantum probe state are erased, using the quantum Fisher information (QFI) we show that the resultant noisy probe can give an estimate of the magnetic field with a precision that scales inversely with the variances of the weight distributions of the corresponding $2^t$ shortened codes. If $C$ is any code concatenated with inner repetition codes of length linear in $n$, a quantum advantage in QM is possible. Hence, given any CSS code of constant length, concatenation with repetition codes of length linear in $n$ is asymptotically optimal for QM with a constant number of erasure errors. We also explicitly construct an observable that when measured on such noisy code-inspired probe states, yields a precision on the magnetic field strength that also exhibits a quantum advantage in the limit of vanishing magnetic field strength. We emphasize that, despite the use of coding-theoretic methods, our results do not involve syndrome measurements or error correction. We complement our results with examples of probe states constructed from Reed-Muller codes.Comment: 21 pages, 3 figure

Finding the optimal probe state for multiparameter quantum metrology using conic programming

The aim of the channel estimation is to estimate the parameters encoded in a quantum channel. For this aim, it is allowed to choose the input state as well as the measurement to get the outcome. Various precision bounds are known for the state estimation. For the channel estimation, the respective bounds are determined depending on the choice of the input state. However, determining the optimal input probe state and the corresponding precision bounds in estimation is a non-trivial problem, particularly in the multi-parameter setting, where parameters are often incompatible. In this paper, we present a conic programming framework that allows us to determine the optimal probe state for the corresponding multi-parameter precision bounds. The precision bounds we consider include the Holevo-Nagaoka bound and the tight precision bound that give the optimal performances of correlated and uncorrelated measurement strategies, respectively. Using our conic programming framework, we discuss the optimality of a maximally entangled probe state in various settings. We also apply our theory to analyze the canonical field sensing problem using entangled quantum probe states.Comment: 36 pages, 2 columns, 5 figures. Title change, added references, and edited introductio

Trade-offs on number and phase shift resilience in bosonic quantum codes

Minimizing the number of particles used by a quantum code is helpful, because every particle incurs a cost. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer $g$ apart, and the gap $g$ imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting $g$-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, $g$-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing errors.Comment: 8 pages, 3 figures, 2 column