13 research outputs found
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which
are approximate quantum error-correcting codes specified as the ground space of
a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum
error-correcting codes with sparse, low-weight stabilizer generators (i.e. each
stabilizer generator acts on a few qubits, and each qubit participates in a few
stabilizer generators). Our investigation is motivated by an important question
in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes
with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo
polylogarithmic corrections) is possible if we go beyond stabilizer codes: we
prove the existence of a family of approximate QLDPC
codes that encode logical qubits into physical
qubits with distance and approximation infidelity
. The code space is
stabilized by a set of 10-local noncommuting projectors, with each physical
qubit only participating in projectors. We
prove the existence of an efficient encoding map, and we show that arbitrary
Pauli errors can be locally detected by circuits of polylogarithmic depth.
Finally, we show that the spectral gap of the code Hamiltonian is
by analyzing a spacetime circuit-to-Hamiltonian
construction for a bitonic sorting network architecture that is spatially local
in dimensions.Comment: 51 pages, 13 figure
Universal Hamiltonians for Exponentially Long Simulation
We construct a Hamiltonian whose dynamics simulate the dynamics of every other Hamiltonian up to exponentially long times in the system size. The Hamiltonian is time-independent, local, one-dimensional, and translation invariant. As a consequence, we show (under plausible computational complexity assumptions) that the circuit complexity of the unitary dynamics under this Hamiltonian grows steadily with time up to an exponential value in system size. This result makes progress on a recent conjecture by Susskind, in the context of the AdS/CFT correspondence, that the time evolution of the thermofield double state of two conformal fields theories with a holographic dual has a circuit complexity increasing linearly in time, up to exponential time
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N,k,d,Δ]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity Δ = 1/(N). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in N projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N^(â3.09)). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.
Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit-to-Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits. Because of this, we call our codes spacetime codes.
The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n^(â3.09)Dâ»ÂČ logâ»â¶ (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth logn, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to 1+1 dimensional nearest-neighbor quantum circuits with at least linear depth
Fragile boundaries of tailored surface codes
Biased noise is common in physical qubits, and tailoring a quantum code to
the bias by locally modifying stabilizers or changing boundary conditions has
been shown to greatly increase error correction thresholds. In this work, we
explore the challenges of using a specific tailored code, the XY surface code,
for fault-tolerant quantum computation. We introduce efficient and
fault-tolerant decoders, belief-matching and belief-find, which exploit
correlated hyperedge fault mechanisms present in circuit-level noise. Using
belief-matching, we find that the XY surface code has a higher threshold and
lower overhead than the square CSS surface code for moderately biased noise.
However, the rectangular CSS surface code has a lower qubit overhead than the
XY surface code when below threshold. We identify a contributor to the reduced
performance that we call fragile boundary errors. These are string-like errors
that can occur along spatial or temporal boundaries in planar architectures or
during logical state preparation and measurement. While we make partial
progress towards mitigating these errors by deforming the boundaries of the XY
surface code, our work suggests that fragility could remain a significant
obstacle, even for other tailored codes. We expect that our decoders will have
other uses; belief-find has an almost-linear running time, and we show that it
increases the threshold of the surface code to 0.937(2)% in the presence of
circuit-level depolarising noise, compared to 0.817(5)% for the more
computationally expensive minimum-weight perfect matching decoder.Comment: 16 pages, 17 figure
Universal Hamiltonians for Exponentially Long Simulation
We construct a Hamiltonian whose dynamics simulate the dynamics of every other Hamiltonian up to exponentially long times in the system size. The Hamiltonian is time-independent, local, one-dimensional, and translation invariant. As a consequence, we show (under plausible computational complexity assumptions) that the circuit complexity of the unitary dynamics under this Hamiltonian grows steadily with time up to an exponential value in system size. This result makes progress on a recent conjecture by Susskind, in the context of the AdS/CFT correspondence, that the time evolution of the thermofield double state of two conformal fields theories with a holographic dual has a circuit complexity increasing linearly in time, up to exponential time
Subspace Correction for Constraints
We demonstrate that it is possible to construct operators that stabilize the
constraint-satisfying subspaces of computational problems in their Ising
representations. We provide an explicit recipe to construct unitaries and
associated measurements for some such constraints. The stabilizer measurements
allow the detection of constraint violations, and provide a route to recovery
back into the constrained subspace. We call this technique ``subspace
correction". As an example, we explicitly investigate the stabilizers using the
simplest local constraint subspace: Independent Set. We find an algorithm that
is guaranteed to produce a perfect uniform or weighted distribution over all
constraint-satisfying states when paired with a stopping condition: a quantum
analogue of partial rejection sampling. The stopping condition can be modified
for sub-graph approximations. We show that it can prepare exact Gibbs
distributions on regular graphs below a critical hardness in
sub-linear time. Finally, we look at a potential use of subspace correction for
fault-tolerant depth-reduction. In particular we investigate how the technique
detects and recovers errors induced by Trotterization in preparing maximum
independent set using an adiabatic state preparation algorithm.Comment: 12 + 4 pages, 6 figure
Building a fault-tolerant quantum computer using concatenated cat codes
We present a comprehensive architectural analysis for a fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated near-term physical parameters for electro-acoustic systems, we perform a detailed error analysis of measurements and gates, including CNOT and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic-state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1,000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits which are intractable for classical supercomputers. Hardware with 32,000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing
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Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes
We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated physical parameters for the hardware, we perform a detailed error analysis of measurements and gates, including cnot and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits, which are currently intractable for classical computers. Hardware with 18 000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing