Square-lattice GKP code

Square-lattice GKP code[1]

Description

Single-mode GKP qudit-into-oscillator CSS code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension.

Codewords can be expressed as equal weight superpositions of coherent states on a rectangular lattice in phase space with spatial period \(2\sqrt{\pi}\). The exact GKP state is non-normalizable, so approximate constructions have to be considered.

The \(q=1\) trivial encoding is spanned by the canonical GKP state or grid state, \begin{align} |GKP\rangle=\sum_{n\in\mathbb{Z}}|x=n\sqrt{2\pi}\rangle~, \tag*{(1)}\end{align} where \(|x\rangle\) are single-mode position states.

Protection

For stabilizer \(\hat{S}_q(2\alpha),\hat{S}_p(2\beta)\), code can correct displacement errors up to \(\alpha/2\) in the \(q\)-direction and \(\beta/2\) at \(p\)-direction. Approximately protects against photon loss errors [2,3], outperforming most other codes designed to explicitly protect against loss [3]. An analytical expression can be derived for the effective logical channel after loss [4]. Very sensitive to dephasing errors [5]. A biased-noise GKP error correcting code can be prepared by choosing \(\alpha\neq \beta\).

Encoding

Dissipative stabilization of finite-energy square-lattice GKP states using stabilizers conjugated by a cooling ([6], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [7,8]. Preparation of approximate square-lattice GKP states has been studied both theoretically and experimentally [2,9–11]. Various damped versions of GKP states are equivalent [12,13], and there exists a Fock-state expansion [14; Appx. A].Two Josephson junctions coupled by a gyrator [15].Periodic driving (a.k.a. Floquet engineering) [16].Approximate GKP states can be prepared using Gaussian operations and photon detectors [17].An optimal-size circuit using ancillary qubits can be used to prepare an approximate GKP state [18]. The size of the circuit is linear in the logarithm of the approximation parameters of the GKP codes.

Gates

By applying square-lattice GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [19].

Decoding

Syndrome measurement can be done by applying a controlled-displacement controlled by an ancilla qubit. The syndrome information can be obtained by measuring the ancilla qubit after controlled-displacement opearation. See Section. 2D in [5].Decoder [20] based on Knill error correction (a.k.a. telecorrection [21]), which is based on teleportation [22,23].Pauli \(X\),\(Y\) and \(Z\) measurements can be performed by measuring \(-\hat{p},\hat{x}-\hat{p}\) and \(\hat{x}\) repectively. If the measurement outcome is closed to an even multiple of \(\sqrt{\pi}\), then the outcome is +1. If the measurement outcome is closed to an odd multiple of \(\sqrt{\pi}\), then the outcome is -1. See Section. 2D in [5].Reinforcement learning decoder that uses only one ancilla qubit [24]. It has been extended to utilize previously measured syndrome information [25].

Fault Tolerance

Clifford gates can be realized by performing linear-optical operations, sympletic transformations and displacements, all of which are Gaussian operations. Pauli gates can be performed using displacement operators. Clifford gates are fault tolerant in the sense that they map bounded-size errors to bounded-size errors [1].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [26].

Realizations

Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group [27,28], followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state [8]. State preparation also realized by Tan group [29]. Universal gate set, including a two-qubit entangling gate, realized by Tan group [30]. State initialization and application to measuring displacements [31].Microwave cavity coupled to superconducting circuits: reduced form of square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [32]. Subsequent paper by Devoret group [24] uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also [33]). See Ref. [34] for another experiment. A feed-forward-free, i.e., fully autonomous protocol has also been implemented by Nord Quantique [35]. Qudit encodings with \(q=3,4\) have been realized, with logical error rates also beyond break even [36].GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [37].Single-qubit \(Z\)-gate has been demonstrated [38] in the single-photon subspace of an infinite-mode space [39], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode. In this context, GKP position-state wavefunctions are called Dirac combs or frequency combs.

Notes

Single-mode GKP states have been introduced in quantum foundations research defining modular conjugate variables [40] and in coherent-state theory associated with the Heisenberg-Weyl group [41,42][43; Sec. 1.5 and 3.2].The basis formed by the code and error states of a single-mode GKP code is known as the Zak basis, Weil-Brezin transform, or \(kq\) representation in condensed-matter physics [44] and signal processing [45; Ch. 1][46; Eq. (1.112)]. Expansion of a function on \(\mathbb{R}\) in terms of this basis is called the Zak transform [45]. The Segal-Bargmann representations of GKP states are the theta functions of the lowest Landau level on a torus [1][15; Sec. V][47; Prop. 6.3] (see also Refs. [48,49]).GKP syndrome extraction can be used for QKD with squeezed states [50].

Cousins

Approximate quantum error-correcting code (AQECC)— Square-lattice GKP codes approximately protect against photon loss [2,3,51].
Rotor code— Because square-lattice GKP error states are parameterized by two modular (i.e., periodic) variables of position and momentum, measuring one of the GKP stabilizers constrains the oscillator Hilbert space into that of a rotor.
\(\mathbb{Z}^n\) hypercubic lattice— GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
Fusion-based quantum computing (FBQC) code— GKP states can be used to perform computation in a fusion-based encoding [52].
Symmetry-protected topological (SPT) code— The Segal-Bargmann representations of GKP states are the theta functions of the lowest Landau level on a torus [1][15; Sec. V][47; Prop. 6.3] (see also Refs. [48,49,53]).
Kitaev current-mirror qubit code— Current-mirror code phase gates utilize ancillary osillators in square-lattice GKP states [54,55].
Zero-pi qubit code— Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states [54,55].
Rotor GKP code— GKP (rotor GKP) codes protect against shifts in linear (angular) degrees of freedom.
Number-phase code— Square-lattice GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [56].
Asymmetric quantum code— GKP code parameters against position and momentum displacements can be tuned by the choice of lattice (e.g., square vs rectangular).
Modular-qudit GKP code— The square-lattice GKP code can be obtained from the modular-qudit code by taking the physical qudit dimension to be infinite [1; Sec. II].
Spin GKP code— Spin-GKP code constructions utilize the Holstein-Primakoff mapping [57] (see also [58]) to convert various expressions for square-lattice GKP states into codes for spin systems.

Member of code lists

Primary Hierarchy

References

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Cite as:

“Square-lattice GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/lattice/gkp.yml.