Flag fault-tolerant error correction with arbitrary distance codes

In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting from $v$ faults in the circuit have weight greater than $v$. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.Comment: 29 pages (18 pages main text), 22 figures, 7 tables. Comments welcome! V3 represents the version accepted to quantu

Lower bounds on the non-Clifford resources for quantum computations

We establish lower-bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks, treating stabilizer operations as free. Our bounds apply to adaptive computations using measurements and an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational tasks. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are close to optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision $\varepsilon$ requires at least $1/7\cdot\log_2(1/\varepsilon) - 4/3$ $T$-states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on $\varepsilon$. Up to multiplicative factors, we optimally lower bound the number of $T$ or $CCZ$ states needed to implement the ubiquitous modular adder and multiply-controlled-$Z$ operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.Comment: 62 page

Beyond the Spin Model Approximation for Ramsey Spectroscopy

Ramsey spectroscopy has become a powerful technique for probing non-equilibrium dynamics of internal (pseudospin) degrees of freedom of interacting systems. In many theoretical treatments, the key to understanding the dynamics has been to assume the external (motional) degrees of freedom are decoupled from the pseudospin degrees of freedom. Determining the validity of this approximation -- known as the spin model approximation -- is complicated, and has not been addressed in detail. Here we shed light in this direction by calculating Ramsey dynamics exactly for two interacting spin-1/2 particles in a harmonic trap. We focus on $s$-wave-interacting fermions in quasi-one and two-dimensional geometries. We find that in 1D the spin model assumption works well over a wide range of experimentally-relevant conditions, but can fail at time scales longer than those set by the mean interaction energy. Surprisingly, in 2D a modified version of the spin model is exact to first order in the interaction strength. This analysis is important for a correct interpretation of Ramsey spectroscopy and has broad applications ranging from precision measurements to quantum information and to fundamental probes of many-body systems

Three-dimensional color code thresholds via statistical-mechanical mapping

Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code on the body-centererd cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D string-like and 2D sheet-like logical operators to be $p^{(1)}_\mathrm{3DCC} \simeq 1.9\%$ and $p^{(2)}_\mathrm{3DCC} \simeq 27.6\%$. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the 4- and 6-body random coupling Ising models.Comment: 4+7 pages, 6 figures, 1 tabl

Kaitiakitanga : Māori experiences, expressions, and understandings : a thesis presented in fulfilment of the requirements for the Doctor of Philosophy at Massey University, Manawatū, Aotearoa New Zealand

The article in Appendix 6 is re-published under a Creative Commons Attribution 4.0 International licence (CC BY 4.0).This doctoral thesis investigates Māori experiences, expressions, and understandings of kaitiakitanga. Kaitiakitanga is understood as a philosophy connected to three realms, the metaphysical, physical, and human realms. Whether the practice of kaitiakitanga has morphed from its customary understanding situated within the metaphysical, physical, and human realms to a human centric practice is a key focus explored in this study. Kaitiakitanga is considered an environmental ethic and concerns Māori responsibilities and obligations regarding land, water, wāhi tapu, and taonga. Kaitiakitanga is often defined to justify the Māori worldview regarding the environment, resource management, and sustainability. There is a solid and growing base of literature concerning Kaitiakitanga as an environment, sustainability, and productivity ethic, which is derived primarily from hapū, iwi, Waitangi Tribunals, and legal contexts. However, further research is needed regarding the human realm and human experiences, expressions, and understandings of kaitiakitanga and its practice. Kawharu (1998) argued in her seminal work on kaitiakitanga, that its implementation is as much about managing people as it is about managing resources and specifically noted that kaitiakitanga is not simply an environmental ethic but a socio-environmental ethic. The human realm is explored throughout this thesis to identify areas to grow and develop kaitiakitanga in ways that enable consistent, accurate, and clear use of this Māori body of knowledge. The study is underpinned by a Kaupapa Māori methodology. Application of Kaupapa Māori requires honouring and privileging Māori people and knowledge and this carried an obligation to apply Māori ways of knowing and being across all areas of this study. Kaupapa Māori principles provided signposts to organise ideas, views, and experiences in a way that carried cultural integrity. The principles also provided the base for organising and theming the information gathered from the kaikōrero. All 24 kaikōrero are Māori and their experiences and knowledge spanned a range of contexts such as mātauranga Māori, education, rongoā, rangatahi, social work, and whānau. Both individual and group hui were held with the kaikōrero. Regarding the human realm, there are several key findings. Whānau and kuia are considered key knowledge holders in the human realm, and they are crucial to maintaining and sustaining kaitiakitanga practices. This study identifies human beings as ‘kaitiaki’, where traditionally kaitiaki were understood only as spiritual beings and guides, and these roles are determined by whānau, hapū, and iwi. Whakapapa and mana whenua is the precursor to being a kaitiaki or carrying kaitiakitanga roles in the human realm. When kaitiakitanga is enacted within the human realm it is not isolated or separate from the metaphysical and physical realm. Te reo, whakaaro, tikanga and mātauranga are critical to the Māori experience, expression and understanding of kaitiakitanga in the human realm. Finally, the practice of kaitiakitanga in the human realm engages all parts of the self which includes the inner being, personality, attributes, values, qualities, emotions, and feelings. This thesis extends on the three realms of kaitiakitanga and adds further layers of discussion related to tikanga and mātauranga. It contributes to the kaitiakitanga literature regarding whānau understandings, taiao, taonga tuku iho and tino rangatiratanga. The use of kaitiakitanga, by who and in what contexts are also presented. New knowledge is highlighted regarding te tuakiritanga, whānau, and kaitiaki understandings and roles. The study reaffirms that the three realms of kaitiakitanga are an inseparable part of its understanding and practice and that, when situated outside of kaupapa that is Māori, kaitiakitanga becomes divorced from its traditional understanding and its underpinning elements, and therefore loses integrity. Traditionally and in contemporary times Māori understand kaitiaki as spiritual beings and guides, this is extended on here by naming people as ‘kaitiaki’. This thesis provides a significant contribution to the human realm of kaitiakitanga or what is described by Forster (2012) as active kaitiaki and by Kawharu (1998) as social spheres. This study affirms kaitiakitanga as an environmental ethic but adds that it is underpinned by Kaupapa Māori whereby whakapapa is a pre-cursor to kaitiakitanga, and kaitiakitanga is led, defined, experienced, and practiced by Māori

Universal transversal gates with color codes - a simplified approach

We provide a simplified, yet rigorous presentation of the ideas from Bomb\'{i}n's paper "Gauge Color Codes" [arXiv:1311.0879v3]. Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide an explicit construction of a family of color codes in arbitrary dimensions and describe some of their crucial properties. Within this framework, we explicitly show how to transversally implement the generalized phase gate $R_n=\text{diag}(1,e^{2\pi i/2^n})$, which deviates from the method in "Gauge Color Codes", allowing an arguably simpler proof. We describe how to implement the Hadamard gate $H$ fault-tolerantly using code switching. In three dimensions, this yields, together with the transversal $CNOT$, a fault-tolerant universal gate set $\{H,CNOT,R_3\}$ without state-distillation.Comment: 13 pages, 6 figure

A comparison of short-term and long-term air pollution exposure associations with mortality in two cohorts in Scotland

Air pollution–mortality risk estimates are generally larger at longer-term, compared with short-term, exposure time scales. We compared associations between short-term exposure to black smoke (BS) and mortality with long-term exposure–mortality associations in cohort participants and with short-term exposure–mortality associations in the general population from which the cohorts were selected. We assessed short-to-medium–term exposure–mortality associations in the Renfrew–Paisley and Collaborative cohorts (using nested case–control data sets), and compared them with long-term exposure–mortality associations (using a multilevel spatiotemporal exposure model and survival analyses) and short-to-medium–term exposure–mortality associations in the general population (using time-series analyses). For the Renfrew–Paisley cohort (15,331 participants), BS exposure–mortality associations were observed in nested case–control analyses that accounted for spatial variations in pollution exposure and individual-level risk factors. These cohort-based associations were consistently greater than associations estimated in time-series analyses using a single monitoring site to represent general population exposure {e.g., 1.8% [95% confidence interval (CI): 0.1, 3.4%] vs. 0.2% (95% CI: 0.0, 0.4%) increases in mortality associated with 10-μg/m3 increases in 3-day lag BS, respectively}. Exposure–mortality associations were of larger magnitude for longer exposure periods [e.g., 3.4% (95% CI: –0.7, 7.7%) and 0.9% (95% CI: 0.3, 1.5%) increases in all-cause mortality associated with 10-μg/m3 increases in 31-day BS in case–control and time-series analyses, respectively; and 10% (95% CI: 4, 17%) increase in all-cause mortality associated with a 10-μg/m3 increase in geometic mean BS for 1970–1979, in survival analysis]. After adjusting for individual-level exposure and potential confounders, short-term exposure–mortality associations in cohort participants were of greater magnitude than in comparable general population time-series study analyses. However, short-term exposure–mortality associations were substantially lower than equivalent long-term associations, which is consistent with the possibility of larger, more persistent cumulative effects from long-term exposures