Magic State Research Articles

Cost-reduced all-Gaussian universality with the Gottesman-Kitaev-Preskill code: Resource-theoretic approach to cost analysis

The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $\lvert 0\rangle$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $\lvert H\rangle$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast, we propose a scheme of preparing $\lvert H\rangle$ directly and combining Gaussian operations only with $\lvert H\rangle$ to achieve the universality without this magic state distillation. In addition, we develop an analytical method to obtain bounds of fundamental limit on transformation between $\lvert H\rangle$ and $\lvert 0\rangle$, finding an application of quantum resource theories to cost analysis of quantum computation with the GKP code. Our results lead to an essential reduction of required non-Gaussian resources for photonic fault-tolerant quantum computation compared to the previous scheme.

Lower bounds on the non-Clifford resources for quantum computations

Treating stabilizer operations as free, we establish lower bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks. Our bounds apply to adaptive computations using measurements with an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational subroutines including the quantum adder and the multiply-controlled Z gate. 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 optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision ɛ requires at least 1/7 ⋅ log2(1/ɛ) − 4/3T-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 ɛ. 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.

Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.

Efficiently Computable Bounds for Magic State Distillation.

Magic-state distillation (or nonstabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to nonstabilizer state manipulation is the resource theory of nonstabilizer states, for which one of the goals is to characterize and quantify nonstabilizerness of a quantum state. In this Letter, we introduce the family of thauma measures to quantify the amount of nonstabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of nonstabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable nonstabilizerness, which in turn leads to a variety of bounds on the rate at which nonstabilizerness can be distilled, as well as on the overhead of magic-state distillation. We then prove that the max-thauma can be used as an efficiently computable tool in benchmarking the efficiency of magic-state distillation, and that it can outperform previous approaches based on mana. Finally, we use the min-thauma to bound a quantity known in the literature as the "regularized relative entropy of magic." As a consequence of this bound, we find that two classes of states with maximal mana, a previously established nonstabilizerness measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of nonstabilizer states and reveals a difference between the resource theory of nonstabilizer states and other resource theories such as entanglement and coherence.

Quantifying the resource content of quantum channels: An operational approach

We propose a general method to operationally quantify the resourcefulness of quantum channels via channel discrimination, an important information processing task. A main result is that the maximum success probability of distinguishing a given channel from the set of free channels by free probe states is exactly characterized by the resource generating power, i.e. the maximum amount of resource produced by the action of the channel, given by the trace distance to the set of free states. We apply this framework to the resource theory of quantum coherence, as an informative example. The general results can also be easily applied to other resource theories such as entanglement, magic states, and asymmetry.

Phase-space-simulation method for quantum computation with magic states on qubits

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.

Contextual bound states for qudit magic state distillation

Identifying necessary and sufficient conditions for universal quantum computing is a long-standing open problem for which contextuality is, perhaps, the only promising solution [Howard et al., Nature 510, 351 (2014)]. To justify this conjecture, Howard et al. showed that contextuality is equivalent to Wigner negativity for qudits, and is therefore necessary and possibly sufficient for qudit magic state distillation. Here, we reformulate magic state distillation in the language of discrete phase space, to show that any finite qudit magic state distillation routine contains bound states outside the Wigner polytope -- these states exhibit contextuality but are useless for magic state distillation.

Repetition Cat Qubits for Fault-Tolerant Quantum Computation

We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardware-efficient universal and fault-tolerant quantum computation. The cat qubits that are stabilized by a two-photon driven-dissipative process, exhibit a tunable noise bias where the effective bit-flip errors are exponentially suppressed with the average number of photons. We propose a realization of a set of gates on the cat qubits that preserve such a noise bias. Combining these base qubit operations, we build, at the level of the repetition cat qubit, a universal set of fully protected logical gates. This set includes single-qubit preparations and measurements, NOT, controlled-NOT, and controlled-controlled-NOT (Toffoli) gates. Remarkably, this construction avoids the costly magic state preparation, distillation, and injection. Finally, all required operations on the cat qubits could be performed with slight modifications of existing experimental setups.

Magic State Distillation: Not as Costly as You Think

Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.

All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code.

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.

Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States.

We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to approximate the output probabilities, with respect to the l_{1} norm, of a large fraction of Clifford circuits. The running time of our algorithm decreases as the inputs become more mixed. Second, we consider the case when the input state is a pure nonstabilizer product state, and show that a similar efficient algorithm exists to approximate the output probabilities, when a suitable restriction is placed on the number of qubits measured. This restriction depends on a magic monotone that we call the Pauli rank. We apply our results to give an efficient output probability approximation algorithm for some restricted quantum computation models, such as Clifford circuits with solely magic state inputs, Pauli-based computation, and instantaneous quantum polynomial time circuits. Finally, we discuss the relationship between Pauli rank and stabilizer rank.

Universal quantum computing with parafermions assisted by a half-fluxon

Braiding of anyons such as Majoranas or parafermions provides only Clifford gates which do not form a universal set of quantum gates. We propose a robust and resource-efficient scheme to perform a non-Clifford gate on a logical qudit encoded in parafermionic zero modes via the Aharonov-Casher effect. This gate can be implemented by moving a half flux quantum around the pair of parafermionic zero modes. The parafermion modes can be realized in a two-dimensional set-up using existing proposals and a half fluxon carrying half flux quantum can be created as a part of a half fluxon/anti-half fluxon pair in a spin-triplet Josephson junction with a dipole defect. With an appropriate bias current pulse, the half fluxon can be braided around the parafermions. Supplementing this gate with the braiding of parafermions provides the avenue for universal quantum computing with parafermions without magic state distillation.

Group Geometrical Axioms for Magic States of Quantum Computing

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

A Straightforward Approach to Multifunctional Graphene

Graphene has been covalently functionalized through a one-pot reductive pathway using graphite intercalation compounds (GICs), in particular KC8 , with three different orthogonally protected derivatives of 4-aminobenzylamine. This novel multifunctional platform exhibits excellent bulk functionalization homogeneity (Hbulk ) and degree of addition while preserving the chemical functionalities of the organic addends through different protecting groups, namely: tert-butyloxycarbonyl (Boc), benzyloxycarbonyl (Cbz) and phthalimide (Pht). We have employed (temperature-dependent) statistical Raman spectroscopy (SRS), X-ray photoelectron spectroscopy (XPS), magic angle spinning solid state 13 C NMR (MAS-NMR), and a characterization tool consisting of thermogravimetric analysis coupled with gas chromatography and mass spectrometry (TG-GC-MS) to unambiguously demonstrate the covalent binding and the chemical nature of the different molecular linkers. This work paves the way for the development of smart graphene-based materials of great interest in biomedicine or electronics, to name a few, and will serve as a guide in the design of new 2D multifunctional materials.

All Pure Fermionic Non-Gaussian States Are Magic States for Matchgate Computations.

Magic states were introduced in the context of Clifford circuits as a resource that elevates classically simulatable computations to quantum universal capability, while maintaining the same gate set. Here we study magic states in the context of matchgate (MG) circuits, where the notion becomes more subtle, as MGs are subject to locality constraints. Nevertheless a similar picture of gate-gadget constructions applies, and we show that every pure fermionic state which is non-Gaussian, i.e., which cannot be generated by MGs from a computational basis state, is a magic state for MG computations. This result has significance for prospective quantum computing implementation in view of the fact that MG circuit evolutions coincide with the quantum physical evolution of noninteracting fermions.

Towards Low Overhead Magic State Distillation.

Magic-state distillation is a resource intensive subroutine for quantum computation. The ratio of noisy input states to output states with an error rate at most ε scales as O(log^{γ}(1/ε)) [S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012)10.1103/PhysRevA.86.052329]. In a breakthrough paper, Hastings and Haah [Distillation with Sublogarithmic Overhead, Phys. Rev. Lett. 120, 050504 (2018)10.1103/PhysRevLett.120.050504] showed that it is possible to construct distillation routines with a sublogarithmic overhead, achieving γ≈0.6779 and falsifying a conjecture that γ is lower bounded by 1. They then ask whether γ can be made arbitrarily close to 0. We answer this question in the affirmative for magic-state distillation routines using qudits of prime dimension (d dimensional quantum systems for prime d).

Unifying the Clifford hierarchy via symmetric matrices over rings

The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (arXiv:1608.06596) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal gates whose action on a computational basis qudit state is described by a $2^k$-th root of unity raised to a polynomial function of the state, and they established the level of such unitaries in the hierarchy. For qubit systems, we consider $k$-th level diagonal gates that can be described just by quadratic forms of the state over the ring $\mathbb{Z}_{2^k}$ of integers mod $2^k$. These involve symmetric matrices over $\mathbb{Z}_{2^k}$ that can be used to efficiently describe all $2$-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal gates from lower levels. This involves symplectic matrices over $\mathbb{Z}_{2^k}$ and hence our perspective unifies these gates with the binary symplectic framework for Clifford gates. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates might be understood by generalizing other binary symplectic matrices to integer rings.

Universal fault-tolerant quantum computation using fault-tolerant conversion schemes

In this paper, we present the fault-tolerant conversion between quantum Reed–Muller (QRM)(2, 5) and QRM(2, 7), and also the conversion between QBCH(15, 7) and QRM(2, 7). Either of the two schemes provides a method to realize universal fault-tolerant quantum computation. In particular, the gate overhead and logical error rate of a logical T gate are provided, as well as the comparison with magic state distillation scheme. In addition, we propose two other fault-tolerant conversion schemes based on and constructions.

One-Shot Operational Quantum Resource Theory.

A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-nonspecific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max- or min-relative entropies and hypothesis testing relative entropy, as well as the free robustness measure, providing them with general operational meanings in terms of optimal state conversion. Our results encompass a wide class of resource theories via the theory-dependent coefficients we introduce, and the discussions are solidified by important examples, such as entanglement, coherence, superposition, magic states, asymmetry, and thermal nonequilibrium.

Three-dimensional surface codes: Transversal gates and fault-tolerant architectures

One of the leading quantum computing architectures is based on the two-dimensional (2D) surface code. This code has many advantageous properties such as a high error threshold and a planar layout of physical qubits where each physical qubit need only interact with its nearest neighbours. However, the transversal logical gates available in 2D surface codes are limited. This means that an additional (resource intensive) procedure known as magic state distillation is required to do universal quantum computing with 2D surface codes. Here, we examine three-dimensional (3D) surface codes in the context of quantum computation. We introduce a picture for visualizing 3D surface codes which is useful for analysing stacks of three 3D surface codes. We use this picture to prove that the $CZ$ and $CCZ$ gates are transversal in 3D surface codes. We also generalize the techniques of 2D surface code lattice surgery to 3D surface codes. We combine these results and propose two quantum computing architectures based on 3D surface codes. Magic state distillation is not required in either of our architectures. Finally, we show that a stack of three 3D surface codes can be transformed into a single 3D color code (another type of quantum error-correcting code) using code concatenation.