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Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi, Poulin & Terhal (PRL 2010) have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d ∝ n 1/2+ε requires Ω(n 1/2+ε ) interactions of length Ω(n ε ). Further a code satisfying k ∝ n with distance d ∝ n α requires Ω(n) interactions of length Ω(n α/2 ). As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.

We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code in 2D at a logical error rate of δ, a space overhead of Ω( log(1/δ)) is needed for any constant depolarizing noise p > 0.

Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi & Terhal, and Bravyi, Poulin & Terhal have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d ∝ n 1/2+ε requires Ω(n 1/2+ε ) interactions of length Ω(n ε ). Further a code satisfying k ∝ n with distance d ∝ n α requires Ω(n) interactions of length Ω(n α/2 ). Our results are derived using bounds on quantum codes from graph metrics. As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.