Magic-state preparation is a central component of fault-tolerant quantum computing. Recent theoretical and experimental successes in code-switch-based magic-state preparation have underscored the promise of these methods for quantum error correction. Similarly, magic-state cultivation has likewise been demonstrated in both numerical and experimental settings. However, a thorough comparison between magic-state cultivation and code-switch-based magic-state factories is still missing. In this work, we carry out end-to-end simulations of magic-state preparation using code switching and compare its resource requirements and performance against magic-state cultivation. As part of this analysis, we develop a lattice-surgery protocol for transfer between the doubled color code and the rotated surface code. We extend the complete code-switching protocol to the $d=5$ doubled color code and perform the corresponding end-to-end simulations. Finally, we propose two fault-tolerant magic-state preparation protocols that combine phase-kickback checks with a transversal non-Clifford gate.
Magic-state preparation is a central component of fault-tolerant quantum computing. Recent theoretical and experimental successes in code-switch-based magic-state preparation have underscored the promise of these methods for quantum error correction. Similarly, magic-state cultivation has likewise been demonstrated in both numerical and experimental settings. However, a thorough comparison between magic-state cultivation and code-switch-based magic-state factories is still missing. In this work, we carry out end-to-end simulations of magic-state preparation using code switching and compare its resource requirements and performance against magic-state cultivation. As part of this analysis, we develop a lattice-surgery protocol for transfer between the doubled color code and the rotated surface code. We extend the complete code-switching protocol to the d=5 doubled color code and perform the corresponding end-to-end simulations. Finally, we propose two fault-tolerant magic-state preparation protocols that combine phase-kickback checks with a transversal non-Clifford gate.
Efficient Magic State Cultivation for T Gates
Recently, experimental and theoretical quantum error correction methodology has seen remarkable breakthroughs. In particular, magic state cultivation has been shown to simplify magic-state preparation and make it feasible for near-term devices. However, recent research on magic state cultivation has focused primarily on the cultivation of T∣+⟩L. Only a few other magic state cultivation methods beyond T∣+⟩L have been investigated. Here, we generalize phase kickback checks for magic states at arbitrary Clifford hierarchy levels in specific codes. We provide an example of cultivation of T∣+⟩L in the doubled color code and the corresponding escape strategy using lattice surgery from the color code to large rotated surface codes. Using state vector simulation for un-grown cultivation, we observe a strong consistence between S∣+⟩L and T∣+⟩L cultivation's performance on the doubled color code. Finally, we discuss the application of the corresponding T∣+⟩L cultivation, incorporating the STAR architecture and T gates, for early fault-tolerant quantum computing and its potential to shorten gate synthesis in the fully fault-tolerant quantum computing era.
Magic state cultivation prepares high-fidelity magic states at low expected space-time costs; however, the exact performance of some schemes is unsettled due to the difficulty in simulating non-Clifford circuits. Here, we use matrix-product states (MPS) based methods to compute the exact performance of two types of fold-transversal cultivation schemes: (i) the Sahay et al method based on the regular surface code S gate, and (ii) a method we propose based on a partially fault-tolerant fold-transversal S gate. We show that for the former protocol at d=5, the ∣T⟩ output reaches similar logical error rates to the ∣S⟩ output, traditionally used as a cheap full Clifford proxy. This contrasts with the ∼10× discrepancy reported for the d=5 colour-code scheme of Gidney et al. We also find that our new d=5 scheme has ∼1.3× lower expected space-time cost while still reaching 10−9 logical error rate. We show that MPS and Clifford-augmented MPS (CAMPS) perform on par with or even better than the recently introduced near-Clifford simulator Clifft on the hardest d=5 regular surface code scheme. Additionally, to speed up simulation, we propose a new pre-screening method based on simple Pauli propagation, lowering by up to three orders of magnitude the required number of exact simulations, and use several simulator-agnostic sampling methods such as subset sampling.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical T gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer Z-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical Z-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of Z-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of r-orthogonal quantum codes, r≥2, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical Z-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of Z-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of O(θL2(1−Θ(1/d))pph), while requiring only the ancillary space of a single surface code patch. Here, θL is the logical rotation angle, pph is the physical error rate, and d is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, O(θLpph), making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for θL≲10−5, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard T-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+T+φ gate set, rather than the conventional Clifford+T or Clifford+φ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of pph=10−3.