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

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

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

Quantum error correction on symmetric quantum sensors

Symmetric states of collective angular momentum are good candidates for multi-qubit probe states in quantum sensors because they are easy to prepare and can be controlled without requiring individual addressability. Here, we give quantum error correction protocols for estimating the magnitude of classical fields using symmetric probe states. To achieve this, we first develop a general theory for quantum error correction on the symmetric subspace. This theory, based on the representation theory of the symmetric group, allows us to construct efficient algorithms that can correct any correctible error on any permutation-invariant code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric pulse gates. For deletion errors, we give a simpler quantum error correction algorithm based on primarily on geometric pulse gates. Second, we devise a simple quantum sensing scheme on symmetric probe states that works in spite of a linear rate of deletion errors, and analyze its asymptotic performance. In our scheme, we repeatedly project the probe state onto the codespace while the signal accumulates. When the time spent to accumulate the signal is constant, our scheme can do phase estimation with precision that approaches the best possible in the noiseless setting. Third, we give near-term implementations of our algorithms.Comment: 26 pages, 7 figures, 2 column

A More Accurate Measurement Model for Fault-Tolerant Quantum Computing

Aliferis, Gottesman and Preskill [1] reduce a non-Markovian noise model to a local noise model, under assumptions on the smallness of the norm of the system-bath interaction. They also prove constructively that given a local noise model, it is possible to simulate an ideal quantum circuit with size L and depth D up to any accuracy, using circuit constructed out of noisy gates from the Boykin set with size $L' = O(L (log L)^a)$ and depth $D'=O(D (log D)^b)$, where $a$ and $b$ are constants that depend on the error correction code that we choose and the design of the fault-tolerant architecture, in addition to more assumptions [1]. These two results combined give us a fault-tolerant threshold theorem for non-Markovian noise, provided that the strength of the effective local noise model is smaller than a positive number that depends on the fault-tolerant architecture we choose. However the ideal measurement process may involve a strong system-bath interaction which necessarily gives a local noise model of large strength. We refine the reduction of the non-Markovian noise model to the local noise model such that this need not be the case, provided that system-bath interactions from the non-ideal operations is sufficiently small. We make all assumptions that [1] has already made, in addition to a few more assumptions to obtain our result. We also give two specific instances where the norm of the fault gets suppressed by some paramater other than the norm of the system-bath interaction. These include the large ratio of the norm of the ideal Hamiltonian to the norm of the perturbation, and frequency of oscillation of the perturbation. We hence suggest finding specific phenomenological models of noise that exhibit these properties