Error-corrected quantum metrology

Quantum metrology, which studies parameter estimation in quantum systems, has many applications in science and technology ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on the estimation precision, called the Heisenberg limit (HL), which bears a quadratic enhancement over the standard quantum limit (SQL) determined by classical statistics. The HL is achievable in ideal quantum devices, but is not always achievable in presence of noise. Quantum error correction (QEC), as a standard tool in quantum information science to combat the effect of noise, was considered as a candidate to enhance quantum metrology in noisy environment. This thesis studies metrological limits in noisy quantum systems and proposes QEC protocols to achieve these limits. First, we consider Hamiltonian estimation under Markovian noise and obtain a necessary and sufficient condition called the ``Hamiltonian-not-in-Lindblad-span\u27\u27 condition to achieve the HL. When it holds, we provide ancilla-assisted QEC protocols achieving the HL; when it fails, the SQL is inevitable even using arbitrary quantum controls, but approximate QEC protocols can achieve the optimal SQL coefficient. We generalize the results to parameter estimation in quantum channels, where we obtain the ``Hamiltonian-not-in-Kraus-span\u27\u27 condition and find explicit formulas for asymptotic estimation precision by showing attainability of previously established bounds using QEC protocols. All QEC protocols are optimized via semidefinite programming. Finally, we show that reversely, metrological bounds also restrict the performance of error-correcting codes by deriving a powerful bound in covariant QEC

Achieving the Heisenberg limit in quantum metrology using quantum error correction

Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed which suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.Comment: 16 pages, 2 figures, see also arXiv:1704.0628

Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies

We systematically study the fundamental competition between quantum error correction (QEC) and continuous symmetries, two key notions in quantum information and physics, in a quantitative manner. Three meaningful measures of approximate symmetries in quantum channels and in particular QEC codes, respectively based on the violation of covariance conditions over the entire symmetry group or at a local point, and the violation of charge conservation, are introduced and studied. Each measure induces a corresponding characterization of approximately covariant codes. We explicate a host of different ideas and techniques that enable us to derive various forms of trade-off relations between the QEC inaccuracy and all symmetry violation measures. More specifically, we introduce two frameworks for understanding and establishing the trade-offs respectively based on the notions of charge fluctuation and gate implementation error, and employ methods including the Knill--Laflamme conditions as well as quantum metrology and quantum resource theory for the derivation. From the perspective of fault-tolerant quantum computing, our bounds on symmetry violation indicate limitations on the precision or density of transversally implementable logical gates for general QEC codes, refining the Eastin--Knill theorem. To exemplify nontrivial approximately covariant codes and understand the achievability of the above fundamental limits, we analyze the behaviors of two explicit types of codes: a parametrized extension of the thermodynamic code (which gives a construction of a code family that continuously interpolates between exact QEC and exact symmetry), and the quantum Reed--Muller codes. We show that both codes can saturate the scaling of the bounds for group-global covariance and charge conservation asymptotically, indicating the near-optimality of these bounds and codes.Comment: 46 pages, 4 figures, long version of arXiv:2111.06355 published as supplementary informatio

Approximate symmetries and quantum error correction

Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin--Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed--Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.Comment: 19 pages, 2 figures, published version, concise version of arXiv:2111.0636