Description
Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\).
The centralizer for the stabilizer group within the displacement operators case can be identified with the symplectic dual lattice \({\mathcal{L}}^{\perp}\) (i.e. all points in \(\mathbb{R}^{2n}\) that have integer symplectic inner product with all points in \({\mathcal{L}}\) ), such that logical operations are identified with the dual quotients \({\mathcal{L}}^{\perp}/{\mathcal{L}}\). The size of this dual quotient is the determinant of the Gram matrix, yielding the logical dimension \(d=\sqrt{\| \det{A}\|}\) [1]. Stabilizer generator matrices equivalent under symplectic transformations are classified by distinct Hermite normal forms [3].
The space of all single-mode GKP codes is the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed [4].
Protection
The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) [3].
Rate
Transmission schemes with multimode GKP codes achieve a lower bound on displacement noise and a lower bound on the thermal-noise Gaussian channel capacities [2,5–7]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [2]. Particular families of GKP codes achieve the capacity of AD and amplifications channels [8].
Encoding
GKP codes with fixed \(n\) and prime-dimensional logical Hilbert space are symplectically related to a disjoint product of single-mode GKP codes on \(n\) modes, such that encoding via Gaussian unitaries is possible.Dissipative stabilization of finite-energy GKP states using stabilizers conjugated by cooling ([9], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [10,11].
Gates
Gaussian operations and homodyne measurements on GKP states are classically simulable, and there is a sufficient condition for an additional element to achieve universal quantum computation [12].There is a relation between magic (i.e., how far away a state is from being a stabilizer state) and non-Gaussianity for GKP codewords [13,14]. In particular, implementing a non-Clifford logical gate requires a higher degree of non-Gaussianity than that expressed by ideal non-normalizable GKP states [14].By applying GKP error correction to Gaussian input states, computational universality can be achieved without additional non-Gaussian elements [15]. This procedure can be alternatively desscribed as performing heterodyne detection on one half of a GKP encoded Bell pair.Logical shadow tomography protocol [16].
Decoding
The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) [17] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code [3,18–20]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [21].Babai's nearest plane algorithm [22] can be used for bounded-distance decoding [21].Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [7].ML decoder for correcting shift errors in GKP two-qubit gates [23].
Fault Tolerance
Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [1]. For single-mode GKP codes, these operations correspond to non-trivial loops in the space of all single-mode GKP codes (the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed) [4]. Such gates provide another example of monodromy under the particular notion of parallel transport introduced in Ref. [24].
Notes
Parent
- Quantum lattice code — GKP codes are \(n\)-mode quantum lattice codes with \(2n\) stabilizers, i.e., constructed using a non-degenerate lattice.
Children
Cousins
- Approximate quantum error-correcting code (AQECC) — Approximate error-correction offered by GKP codes yields achievable rates that are a constant away from the capacity of Guassian loss channels [2,5–7].
- Amplitude-damping (AD) code — Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of AD, displacement-noise, and thermal-noise Gaussian loss channels [2,5–7]. Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [7].
- Quadrature-amplitude modulation (QAM) code — Finite-energy GKP codes are quantum analogues of lattice-based QAM codes in that both use a subset of points on a lattice.
- Numerically optimized bosonic code — Numerically optimizing GKP code lattices yields codes for three, seven, and nine modes with larger distances and fidelities than known GKP codes [21]. Neural networks can be used to optimize approximate GKP states [29].
References
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Page edit log
- Jonathan Conrad (2022-07-05) — most recent
- Victor V. Albert (2022-07-05)
- Victor V. Albert (2022-03-24)
Cite as:
“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp