Combining fault tolerance and strictly unitary simulation in a universal model of quantum computation based on topological modular functors

Freedman, Larsen, and Wang (Commun. Math. Phys. 227:605-622, 2002) proved that the topological modular functor of SU(2) Witten-Chern-Simons theory at a fifth root of unity is universal for quantum computation, and defined two complementary computational models: CS5, which is fault tolerant but uses intermediate non-unitary label measurements (Hermitian operators $H_\gamma$) to repair a novel 'qubit smearing' (leakage) error, and ECS5, which is strictly unitary (braids act exactly, measurements occur only at the very end) but confers no fault tolerance beyond intrinsic topological protection. The paper explicitly leaves open whether both properties can be achieved simultaneously: 'It is an interesting open problem whether fault tolerance and strictly unitary can be combined in a universal model of computation based on topological modular functors.' No resolution is known to date.

Background

Topological quantum computation (TQC) proposes to store and process quantum information in the global, topological degrees of freedom of certain two-dimensional quantum systems, so that local noise cannot easily corrupt the computation. In Kitaev's formulation (Ann. Phys. 303:2-30, 2003), topologically ordered media support emergent quasiparticles - anyons - whose exchange operations are represented by unitary matrices acting on a degenerate ground-state space; braiding anyons around one another performs quantum gates that are intrinsically ('passively') protected because the information is encoded non-locally.

The mathematical backbone of anyonic computation is the modular functor of a topological quantum field theory (TQFT, a 3-dimensional field theory with no local propagating degrees of freedom, whose observables depend only on topology). A topological modular functor (TMF) assigns to each closed oriented surface $\Sigma$ carrying labeled punctures a finite-dimensional Hilbert space $V(\Sigma)$, and to each diffeomorphism of the surface a unitary map of $V(\Sigma)$; for a disk with $n$ marked points this yields a unitary representation of the braid group $B_n$ (the group of motions of $n$ points in the disk, generated by half-twists $\sigma_i$). For the SU(2) Witten-Chern-Simons theory at a primitive $r$-th root of unity $q=e^{2\pi i/r}$, the label set is finite, and the braid-group action coincides with the irreducible sectors of the Jones representation - the representation discovered by Vaughan Jones in operator algebra, whose weighted trace yields the Jones polynomial of the braid's closure link; the sectors are indexed by two-row Young diagrams such as $[3,3]$ and $[4,2]$.

Freedman, Larsen, and Wang (Commun. Math. Phys. 227:605-622, 2002) proved that the SU(2) modular functor at $q=e^{2\pi i/5}$ (label set {0,1,2,3}) is universal for quantum computation: any polynomial-size quantum circuit - i.e., any computation in BQP, the class of problems efficiently solvable with bounded error on a quantum computer - can be efficiently approximated by braids acting on the functor's state spaces. The technical heart is a density theorem (Theorem 4.1 of the paper): the image of $\rho_{[3,3]}\oplus\rho_{[4,2]}:B_6\to U(5)\times U(8)$ is dense in $SU(5)\times SU(8)$ up to phases, which, combined with a Solovay-Kitaev-type compilation (a word of length $O(1/\epsilon^2)$ in the braid generators approximates any desired gate to accuracy $\epsilon$), lets braids simulate one- and two-qubit gates.

To turn density into a computational model, the paper encodes each qubit in the space $V(D;\,3\text{ punctures})\cong\mathbb{C}^2$ of a disk with three punctures of label 1 ($k$ qubits correspond to $3k$ punctures). Two coupled error phenomena must be handled: ordinary decoherence, and a novel 'qubit smearing' - the braid image approximates the intended gate only up to leakage of weight into complementary (non-computational) sectors of the functor. The model CS5 ('Chern-Simons at a 5th root of unity') repairs smearing by an intermediate non-unitary step: after each simulated gate one measures the topological label on a curve $\gamma=\partial D_i$ using a hypothesized Hermitian operator $H_\gamma$, projecting the state back into the computational subspace; a rare bad outcome (label 3) erases one qubit, which is replaced from a reservoir of 'fresh ancilli' (freshly prepared three-puncture qubits) - the entropy-disposal resource that, as the authors note, any unbounded fault-tolerant computation requires and whose realization is 'inherently not unitary' in ordinary architectures. Assuming $H_\gamma$ and the readout observables $\sigma_z^i$ (Pauli $z$-measurements on the encoded qubits) can be effected in constant or poly time, the paper's Theorem 2.2 shows CS5 efficiently and fault-tolerantly simulates the quantum circuit model, by importing the threshold framework of Aharonov and Ben-Or for circuits with small systematic errors plus rare erasures (SIAM J. Comput. 38:1207-1282). The paper also defines the variant ECS5 ('exact CS5'), in which all braid actions are assumed to act exactly and $H_\gamma$-measurement is performed only once, at the very end, before the $\sigma_z$ readout; Theorem 2.3 gives an efficient and strictly unitary simulation of the quantum circuit model by ECS5. But under ECS5's exactness assumption the braids make no errors, so there is nothing for intermediate measurements to correct: the model 'cannot confer additional fault tolerance', i.e., no protection beyond what the topology itself might passively provide. The authors conclude with the question that is the subject of this problem record: 'It is an interesting open problem whether fault tolerance and strictly unitary can be combined in a universal model of computation based on topological modular functors.' The question is the natural sharpening of the earlier problem of Freedman, Kitaev, and Wang (Commun. Math. Phys. 227:587-603, 2002), who asked of the unitary TMF (UTMF) simulation converse: 'Can some UTMF efficiently simulate a universal quantum computer? Fault tolerantly? We would conjecture the answer is yes to both questions.'

Problem Statement

Does there exist a universal model of computation based on a topological modular functor that is simultaneously (i) strictly unitary - the evolution of the computational state, from its initial preparation to just before the final readout, is a unitary action on the functor's state spaces (braid words / mapping-class-group actions), with no intermediate non-unitary operations such as the CS5 model's mid-computation label measurements $H_\gamma$ (the Hermitian operators with which the source's measurement-assisted model projects the state back into the computational subspace by reading the topological label on a curve encircling a qubit's punctures), ancilla resets or replacements, postselection, or other measurement-and-correction steps - and (ii) fault tolerant beyond the intrinsic, passive protection of the topological theory, i.e., accompanied by a proved tolerance guarantee of the Aharonov-Ben-Or threshold type (bounded overall failure probability for computations of arbitrary length, whenever the physical error rates stay below a fixed constant threshold) against errors occurring during the computation (such as imperfectly implemented braids or thermally generated anyons)? This is the open problem stated verbatim by Freedman-Larsen-Wang (2002): 'It is an interesting open problem whether fault tolerance and strictly unitary can be combined in a universal model of computation based on topological modular functors.' The sought model is required to be universal in the source's sense - efficient simulation of the quantum circuit model (BQP), with polynomial overhead - strictly unitary at every mid-computation step, and backed by a quantified fault-tolerance guarantee against a stated error model, where the guarantee protects against errors the topology alone does not already exclude. The problem retains the source's generality: any topological modular functor and any encoding qualify (not only SU(2) Chern-Simons at a fifth root of unity), and the fault-tolerance requirement means protection beyond whatever passive protection the topology itself confers.

References

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