Unifying Gate Synthesis and Magic State Distillation

Abstract

The leading paradigm for performing computation on quantum memories can be encapsulated as distill-thensynthesize. Initially, one performs several rounds of distillation to create high-fidelity magic states that provide one good T gate, an essential quantum logic gate. Subsequently, gate synthesis intersperses many T gates with Clifford gates to realise a desired circuit. We introduce a unified framework that implements one round of distillation and multi-qubit gate synthesis in a single step. Typically, our method uses the same number of T -gates as conventional synthesis, but with the added benefit of quadratic error suppression. Because of this, one less round of magic state distillation needs to be performed, leading to significant resource savings.Development of quantum computers has intensified, spurred on by the prospect that fully fault-tolerant devices are within reach. A major impetus has been new theoretical advances showing practical designs of fault-tolerant devices can tolerate up to one percent noise [1]. The topological surface code or toric code is the most widely known breakthrough, which allows for a robust storage of quantum information. Augmenting the surface code from a static memory to a computer requires additional information processing gadgets. Fault-tolerant information processing can be achieved by a two-step process. In the first step, logical qubits are distilled from noisy resources into high-fidelity magic states [2]. Each magic state can provide a fault-tolerant T -gate, also known as a π/8 phase gate. In the second step, gate-synthesis techniques decompose any desired unitary into a sequence of many T -gates interspersed with Clifford gates. This approach to processing quantum information can be paraphrased as distill-then-synthesize. Most leading laboratories are following designs [3][4][5] within this paradigm of distill-thensynthesize combined with surface codes. While alternative ideas to magic state distillation exist [6][7][8][9], so far they lack the appealing high tolerance to noise [10,11]. We propose a framework where both distillation and synthesis occur simultaneously, which we call synthillation.Fault-tolerance protocols come with a price-tag, an overhead of extra qubits. Consequently, genuinely useful applications may need millions or billions of physical qubits. Improved protocols for magic state distillation [12][13][14] and gate synthesis [15][16][17][18][19][20][21] have reduced resource overheads, but the cost remains formidable and further overhead reduction is extremely valuable. Notable is the Bravyi-Haah magic state distillation (BHMSD) protocol [13] that converts 3k + 8 magic states into k magic states with quadratic error suppression. For large computations, with between 10 10 and 10 15 logical operations, the required precision can be reached by concatenating BHMSD two or three times, assuming an initial physical error rate of order ∼ 0.1%. Multilevel distillation is an effective tool when many rounds are required [14]. Gate synthesis has advanced on tw...

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