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A fault-tolerant quantum processor may be configured using stationary qubits interacting only with their nearest neighbours, but at the cost of significant overheads in physical qubits per logical qubit. Such overheads could be reduced by coherently transporting qubits across the chip, allowing connectivity beyond immediate neighbours. Here we demonstrate high-fidelity coherent transport of an electron spin qubit between quantum dots in isotopically-enriched silicon. We observe qubit precession in the inter-site tunnelling regime and assess the impact of qubit transport using Ramsey interferometry and quantum state tomography techniques. We report a polarization transfer fidelity of 99.97% and an average coherent transfer fidelity of 99.4%. Our results provide key elements for high-fidelity, on-chip quantum information distribution, as long envisaged, reinforcing the scaling prospects of silicon-based spin qubits., Nature Communications, 12 (1), ISSN:2041-1723

Vast numbers of qubits will be needed for large-scale quantum computing due to the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes as well as the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around one hundred logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches., 14 pages, 6 figures, comments welcome; v2 - final author version, revisions made in response to referee reports

The robust generation and manipulation of entangled multiphoton states on-chip has an essential role in quantum computation and communication. Lattice topology has emerged as a means of protecting photonic states from disorder but entanglement across different topologies remained unexplored. We report biphoton entanglement between topologically distinct spatial modes in a bipartite array of silicon waveguides. The results highlight topology as an additional degree of freedom for entanglement and open avenues for investigating information teleportation between trivial and topological modes.

A fault-tolerant quantum computer will be supported by a classical decoding system interfacing with quantum hardware to perform quantum error correction. It is important that the decoder can keep pace with the quantum clock speed, within the limitations on communication that are imposed by the physical architecture. To this end we propose a local `pre-decoder', which makes greedy corrections to reduce the amount of syndrome data sent to a standard matching decoder. We study these classical overheads for the surface code under a phenomenological phase-flip noise model with imperfect measurements. We find substantial improvements in the runtime of the global decoder and the communication bandwidth by using the pre-decoder. For instance, to achieve a logical failure probability of $f = 10^{-15}$ using qubits with physical error rate $p = 10^{-3}$ and a distance $d=22$ code, we find that the bandwidth cost is reduced by a factor of $1000$, and the time taken by a matching decoder is sped up by a factor of $200$. To achieve this target failure probability, the pre-decoding approach requires a $50\%$ increase in the qubit count compared with the optimal decoder., Comment: 12 pages, 8 figures, comments welcome

For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where non-locality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality., Comment: 15+10 pages, 3 figures

The performance requirements for fault-tolerant quantum computing are very stringent. Qubits must be manipulated, coupled, and measured with error rates well below 1%. For semiconductor implementations, silicon quantum dot spin qubits have demonstrated average single-qubit Clifford gate error rates that approach this threshold, notably with error rates of 0.14% in isotopically enriched $^{28}$Si/SiGe devices. This gate performance, together with high-fidelity two-qubit gates and measurements, is only known to meet the threshold for fault-tolerant quantum computing in some architectures when assuming that the noise is incoherent, and still lower error rates are needed to reduce overhead. Here we experimentally show that pulse engineering techniques, widely used in magnetic resonance, improve average Clifford gate error rates for silicon quantum dot spin qubits to 0.043%,a factor of 3 improvement on previous best results for silicon quantum dot devices. By including tomographically complete measurements in randomised benchmarking, we infer a higher-order feature of the noise called the unitarity, which measures the coherence of noise. This in turn allows us to theoretically predict that average gate error rates as low as 0.026% may be achievable with further pulse improvements. These fidelities are ultimately limited by Markovian noise, which we attribute to charge noise emanating from the silicon device structure itself, or the environment.

The ability to measure long-range entanglement may enable robust quantum memory

We report our experimental demonstration of biphoton entanglement between topologically-distinct modes in a bipartite silicon photonics lattice. These results highlight topology as a degree of freedom for entanglement and could have implications in quantum information.

The quantum logic gates used in the design of a quantum computer should be both universal, meaning arbitrary quantum computations can be performed, and fault-tolerant, meaning the gates keep errors from cascading out of control. A number of no-go theorems constrain the ways in which a set of fault-tolerant logic gates can be universal. These theorems are very restrictive, and conventional wisdom holds that a universal fault-tolerant logic gate set cannot be implemented natively, requiring us to use costly distillation procedures for quantum computation. Here, we present a general framework for universal fault-tolerant logic with stabiliser codes, together with a no-go theorem that reveals the very broad conditions constraining such gate sets. Our theorem applies to a wide range of stabiliser code families, including concatenated codes and conventional topological stabiliser codes such as the surface code. The broad applicability of our no-go theorem provides a new perspective on how the constraints on universal fault-tolerant gate sets can be overcome. In particular, we show how non-unitary implementations of logic gates provide a general approach to circumvent the no-go theorem, and we present a rich landscape of constructions for logic gate sets that are both universal and fault-tolerant. That is, rather than restricting what is possible, our no-go theorem provides a signpost to guide us to new, efficient architectures for fault-tolerant quantum computing., 11 pages + 11 page appendix, 8 figures; v2 improvements to presentation

We analyze a readout scheme for Majorana qubits based on dispersive coupling to a resonator. We consider two variants of Majorana qubits: the Majorana transmon and the Majorana box qubit. In both cases, the qubit-resonator interaction can produce sizeable dispersive shifts in the MHz range for reasonable system parameters, allowing for submicrosecond readout with high fidelity. For Majorana transmons, the light-matter interaction used for readout manifestly conserves Majorana parity, which leads to a notion of quantum nondemolition (QND) readout that is stronger than for conventional charge qubits. In contrast, Majorana box qubits only recover an approximately QND readout mechanism in the dispersive limit where the resonator detuning is large. We also compare dispersive readout to longitudinal readout for the Majorana box qubit. We show that the latter gives faster and higher fidelity readout for reasonable parameters, while having the additional advantage of being manifestly QND, and so may prove to be a better readout mechanism for these systems.

Electron spins in semiconductor quantum dots have been intensively studied for implementing quantum computation and high-fidelity single- and two-qubit operations have recently been achieved. Quantum teleportation is a three-qubit protocol exploiting quantum entanglement and it serves as an essential primitive for more sophisticated quantum algorithms. Here we demonstrate a scheme for quantum teleportation based on direct Bell measurement for a single-electron spin qubit in a triple quantum dot utilizing the Pauli exclusion principle to create and detect maximally entangled states. The single spin polarization is teleported from the input qubit to the output qubit. We find this fidelity is primarily limited by singlet–triplet mixing, which can be improved by optimizing the device parameters. Our results may be extended to quantum algorithms with a larger number of semiconductor spin qubits.

Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code -- the XZZX code -- offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation., 10+6 pages, 11 figures; v2 - minor clarifying changes; v3 - final author version. Data and code used to produce our results available at https://bitbucket.org/qecsim/qsdxzzx/. The available software extends and uses services from the quantum error correction simulation package qecsim

Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality., Comment: 22 pages + appendices, 25 figures; v2 improvements to presentation; v3 published version

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally., 39 pages, 16 figures, comments welcome; v2 includes much more explicit detail on the main example model, including boundary conditions and implementations of logical operators through local moves; v3 published version

Topologically ordered materials may serve as a platform for new quantum technologies such as fault-tolerant quantum computers. To fulfil this promise, efficient and general methods are needed to discover and classify new topological phases of matter. We demonstrate that deep neural networks augmented with external memory can use the density profiles formed in quantum walks to efficiently identify properties of a topological phase as well as phase transitions. On a trial topological ordered model, our method's accuracy of topological phase identification reaches 97.4%, and is shown to be robust to noise on the data. Furthermore, we demonstrate that our trained DNN is able to identify topological phases of a perturbed model and predict the corresponding shift of topological phase transitions without learning any information about the perturbations in advance. These results demonstrate that our approach is generally applicable and may be used to identify a variety of quantum topological materials., 13 pages, 5 figures and 4 tables

Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalises to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of $6\%$ for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of $\sim 5\%$ in an experimentally relevant regime where dephasing errors occur at a rate one hundred times greater than bit-flip errors., Comment: 6 pages, 4 figures, comments welcome; v2 published version, now includes supplemental material with a technical description of the implemented decoders

Electron spins in quantum dots are a promising building block for quantum computing. While fundamental gate operations on single and two qubits have been demonstrated, quantum algorithms utilizing three or more qubits have not tested so far. Quantum teleportation is one of the important milestones that can be implemented with three qubits, because it explicitly makes use of quantum entanglement and is applicable to quantum communication and quantum computing. The key of Quantum teleportation is to prepare one of maximally entangled states and perform projection measurement onto them. These can be, in principle, implemented by using a two-qubit gate operation and a series of single-qubit operations. However, realization of those controls remains challenging in three-spin-qubit systems. Instead, the Pauli spin blockade can be utilized to prepare and selectively measure one of the maximally entangled state. This significantly simplifies partial implementation of the quantum teleportation protocol.

Fast, high-fidelity single and two-qubit gates are essential to building a viable quantum information processor, but achieving both in the same system has proved challenging for spin qubits. We propose and analyze an approach to perform a long-distance two-qubit controlled phase (CPHASE) gate between two singlet-triplet qubits using an electromagnetic resonator to mediate their interaction. The qubits couple longitudinally to the resonator, and by driving the qubits near the resonator's frequency they can be made to acquire a state-dependent geometric phase that leads to a CPHASE gate independent of the initial state of the resonator. Using high impedance resonators enables gate times of order 10 ns while maintaining long coherence times. Simulations show average gate fidelities of over 96% using currently achievable experimental parameters and over 99% using state-of-the-art resonator technology. After optimizing the gate fidelity in terms of parameters tuneable in-situ, we find it takes a simple power-law form in terms of the resonator's impedance and quality and the qubits' noise bath., Comment: 11 pages, 2 figures

We investigate the spin of a multielectron GaAs quantum dot in a sequence of nine charge occupancies, by exchange coupling the multielectron dot to a neighboring two-electron double quantum dot. For all nine occupancies, we make use of a leakage spectroscopy technique to reconstruct the spectrum of spin states in the vicinity of the interdot charge transition between a single- and a multielectron quantum dot. In the same regime we also perform time-resolved measurements of coherent exchange oscillations between the single- and multielectron quantum dot. With these measurements, we identify distinct characteristics of the multielectron spin state, depending on whether the dot's occupancy is even or odd. For three out of four even occupancies we do not observe any exchange interaction with the single quantum dot, indicating a spin-0 ground state. For the one remaining even occupancy, we observe an exchange interaction that we associate with a spin-1 multielectron quantum dot ground state. For all five of the odd occupancies, we observe an exchange interaction associated with a spin-1/2 ground state. For three of these odd occupancies, we clearly demonstrate that the exchange interaction changes sign in the vicinity of the charge transition. For one of these, the exchange interaction is negative (i.e. triplet-preferring) beyond the interdot charge transition, consistent with the observed spin-1 for the next (even) occupancy. Our experimental results are interpreted through the use of a Hubbard model involving two orbitals of the multielectron quantum dot. Allowing for the spin correlation energy (i.e. including a term favoring Hund's rules) and different tunnel coupling to different orbitals, we qualitatively reproduce the measured exchange profiles for all occupancies., Comment: 20 pages, 13 figures, 2 tables

Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on 6-leg infinite-length cylinders, and scrutinize the properties of the detected phases. We compare these results with those of the corresponding nearest neighbor model. Our results suggest that, for such long-range Hamiltonians, the long-range quantum fluctuations always lead to long-range correlations, where correlators exhibit power-law decays instead of the conventional exponential drops observed for short-range correlated gapped phases. Our results are relevant for comparisons with recent ion-trap quantum simulator experiments that demonstrate highly-controllable long-range spin couplings for several hundred ions., 16 pages, 12 figures, and no supplemental materials. v4: equivalent to the published version, except a few minor typographical and notation corrections

Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabiliser codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including colour codes and toric codes, as well as more general codes such as abelian quantum double models and codes with fermionic excitations in more than two dimensions., 24 pages, 17 figures, comments welcome; v2 includes various minor improvements; v3 published version

Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate non-linear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality - the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for state-independent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit measurement-based computations are of the same underlying nature, avoiding the pathologies associated with qubit contextuality., Comment: 14 pages, 3 figures, comments welcome; v3 published version

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits., Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new results on the performance of surface codes in the finite-bias regime, obtained with beveled surface codes and an improved tensor network decoder; v3 published version

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state.

We propose and analyse a scheme for performing a long-range entangling gate for qubits encoded in electron spins trapped in semiconductor quantum dots. Our coupling makes use of an electrostatic interaction between the state-dependent charge configurations of a singlet-triplet qubit and the edge modes of a quantum Hall droplet. We show that distant singlet-triplet qubits can be selectively coupled, with gate times that can be much shorter than qubit dephasing times and faster than decoherence due to coupling to the edge modes. Based on parameters from recent experiments, we argue that fidelities above 99% could in principle be achieved for a two-qubit entangling gate taking as little as 20 ns., 14 pages, 7 figures, comments welcome; v2 published version

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 3D cluster state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a global generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high error tolerance of this 3D cluster state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry., 42 pages, 10 figures, comments welcome; v2 published version

We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli Z errors occur more frequently than X or Y errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is in fact at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of Y instead of Z around the faces, as this doubles the number of useful syndrome bits associated with the dominant Z errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes., Comment: 6 pages, 5 figures, comments welcome; v2 includes minor improvements to the numerical results, additional references, and an extended discussion; v3 published version (incorporating supplementary material into main body of paper)

We demonstrate that small quantum memories, realized via quantum error correction in multi-qubit devices, can benefit substantially by choosing a quantum code that is tailored to the relevant error model of the system. For a biased noise model, with independent bit and phase flips occurring at different rates, we show that a single code greatly outperforms the well-studied Steane code across the full range of parameters of the noise model, including for unbiased noise. In fact, this tailored code performs almost optimally when compared with 10,000 randomly selected stabilizer codes of comparable experimental complexity. Tailored codes can even outperform the Steane code with realistic experimental noise, and without any increase in the experimental complexity, as we demonstrate by comparison in the observed error model in a recent 7-qubit trapped ion experiment., Comment: 6 pages, 2 figures, supplementary material; v2 published version

Randomized benchmarking is routinely used as an efficient method for characterizing the performance of sets of elementary logic gates in small quantum devices. In the measurement-based model of quantum computation, logic gates are implemented via single-site measurements on a fixed universal resource state. Here we adapt the randomized benchmarking protocol for a single qubit to a linear cluster state computation, which provides partial, yet efficient characterization of the noise associated with the target gate set. Applying randomized benchmarking to measurement-based quantum computation exhibits an interesting interplay between the inherent randomness associated with logic gates in the measurement-based model and the random gate sequences used in benchmarking. We consider two different approaches: the first makes use of the standard single-qubit Clifford group, while the second uses recently introduced (non-Clifford) measurement-based 2-designs, which harness inherent randomness to implement gate sequences., 10 pages, 4 figures, comments welcome; v2 published version

We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of $c=1$ conformal field theories (CFTs) of two possible types: the ${S}^{1}$ free boson and its ${Z}_{2}$ orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the ${S}^{1}$ boson CFT, and the Ashkin-Teller model with the ${Z}_{2}$-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.