Abstract
A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task. By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory. A feature dubbed quantum speedup. However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup. Here we develop a fault-tolerant version of one family of these sampling problems, which we show can be implemented using quantum circuits of constant depth. We present two constructions, each taking $poly(n)$ physical qubits, some of which are prepared in noisy magic states. The first of our constructions is a constant depth quantum circuit composed of single and two-qubit nearest neighbour Clifford gates in four dimensions. This circuit has one layer of interaction with a classical computer before final measurements. Our second construction is a constant depth quantum circuit with single and two-qubit nearest neighbour Clifford gates in three dimensions, but with two layers of interaction with a classical computer before the final measurements. For each of these constructions, we show that there is no classical algorithm which can sample according to its output distribution in $poly(n)$ time, assuming two standard complexity theoretic conjectures hold. The noise model we assume is the so-called local stochastic quantum noise. Along the way, we introduce various new concepts such as constant depth magic state distillation (MSD), and constant depth output routing, which arise naturally in measurement based quantum computation (MBQC), but have no constant-depth analogue in the circuit model.
Highlights
Quantum computers promise incredible benefits over their classical counterparts in various areas, from breaking RivestShamir-Adleman (RSA) encryption [1], to machine learning [2], to improvements to generic search [3], among others [4,5]
Each taking poly(m) physical qubits, some of which are prepared in noisy magic states
We show that performing O[n3log(n)] copies of zMSD circuits, each of which is composed of O[log(n)] logical qubits as seen in Appendix B2, guarantees with high probability psucc that at least O(n2) copies of zMSD will be successful
Summary
Quantum computers promise incredible benefits over their classical counterparts in various areas, from breaking RivestShamir-Adleman (RSA) encryption [1], to machine learning [2], to improvements to generic search [3], among others [4,5]. Our sampling problems are built on a family of schemes essentially based on local measurements on regular graph states, which correspond to constant depth two-dimensional (2D) nearest-neighbor (NN) quantum circuits showing quantum speedup [14,15,16,18,23,38] We show that these can be made fault tolerant in a way which maintains constant depth of the quantum circuits, albeit with large (but polynomial) overhead in the number of ancilla systems used, and at most two rounds of (efficient) classical computation during the running of the circuit. The authors of [41] presented a fault-tolerant construction of a sampling problem which cannot be performed by any polynomial time classical algorithm, conditioned on three complexity theoretic conjectures holding This consisted of a constant depth quantum circuit, with a non-Clifford component, obtained by using defect-based topological quantum computing [40]. Note that in our 3D NN architecture we use different (fixed) measurement angles to those in [41] to construct a different sampling problem having an anticoncentration property [16,17,38]
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