Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes (2024)

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  Figure 1

  Qubit encoded in a spin using spin-cat states. (a) The Bloch sphere for the qubit encoded in a spin. The two spin-coherent states (stretched states) are the computational basis states, lying on the $Z$ axis and the spin-cat states then lie along the $X$ axis. The spin Wigner function of the states is shown and its strong negativity indicates that spin cats are highly nonclassical. (b) The spin-cat encoding of a qubit in spin $J=9/2$, $d=2J+1=10$ levels. The correctable errors divide the qudit into two subspaces, $\overline{0}$ and $\overline{1}$, shown as blue and purple boxes, respectively. One physical error channel is optical pumping, corresponding to the absorption of photons (blur arrows) followed by spontaneous emission (wavy red arrows), which can lead to amplitude damping.

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  Figure 2

  Protocol for implementing a rank-preserving cnot gate in neutral atomic ${}^{87}\mathrm{Sr}$ based of optimal control and the Rydberg blockade. The spin-cat qubit is encoded in the nuclear spin, $I=F=9/2$, in the electric ground state, $5s^2{}^1{S}_0$. (a) Detailed level diagram and protocol; (b) high-level schematic. When a gate is to be performed, the qudit is excited from the ground-state memory to the long-lived auxiliary metastable state, $5s5p{}^3{P}_2$, $F=9/2$. Entangling interactions occur through excitation from the auxiliary state to the Rydberg state, $5s60s{}^3{S}_1$, $F=11/2$. The error-correctable subspaces, $\overline{0}$ and $\overline{1}$, are represented by blue and purple colored boxes, respectively, in the ground ($G$), auxiliary ($A$), and Rydberg ($R$) manifolds. The gate is performed in four steps. Step I: using quantum optimal control, the population from the ground state is transferred to the auxiliary state while preserving coherence between magnetic sublevels. Each two-level resonance, $|G,M_F⟩\to |A,M_F⟩$, has a detuning ${\mathrm{\Delta }}_{A,M_F}$ and Rabi frequency ${\mathrm{\Omega }}_{A,M_F}$. For the control atom, we promote only the population from the $\overline{1}$ subspace, whereas for the case of the target atom, we promote the population from both the $\overline{0}$ and $\overline{1}$ subspaces to the auxiliary state (see the main text for details). Step II: using $\pi$-polarized light, local addressing, and quantum control, transfer the population from the auxiliary to Rydberg states only for the control atom. Step III: apply the same pulse to the target atom. Due to the Rydberg blockade, this population transfer only occurs when the control atom is in $\overline{0}$ subspace; for the $\overline{1}$ subspace the population is otherwise blockaded. Step IV: using global rf-phase-modulated optimal control, we perform the SU(2) rotation $X=\exp (-i\pi F_x)$ in the auxiliary manifold and simultaneously the identity operator in the Rydberg manifold. The result is a cnot gate—if the control atom is in $\overline{1}$ subspace we apply an $X$ gate to the target atom if the control atom is in $\overline{0}$ subspace we implement an identity operator $\mathbb{1}$. Finally, we will transfer all the states back to the ground state by reversing steps III–I, thus implementing a rank-preserving cnot gate for the spin-cat encoding.

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  Figure 3

  Examples of control waveforms that achieve the transfer of populations between spin manifolds while preserving the coherence between magnetic sublevels. Based on Hamiltonian Eq.(30), we modulate the lasers’ amplitude, detuning, and phase, as piecewise constant functions of time. Using the GRAPE optimal control we find the target isometries. (a) The waveform that implements $V_{\mathrm{tar}}^{(\mathrm{C})}$, which transfer population from ${\overline{1}}_G$ subspace to ${\overline{1}}_A$ subspace while the population in the ${\overline{0}}_G$ subspace is unchanged. (b) The waveform that implements $V_{\mathrm{tar}}^{(\mathrm{T})}$, which transfer population from ${\overline{1}}_G$ subspace to ${\overline{1}}_A$ subspace and ${\overline{0}}_g$ subspace to ${\overline{0}}_A$ subspace. (c) The waveform that implements $V_{\mathrm{tar}}^{(\mathrm{Ryd})}$ that transfers the population from the auxiliary states to the Rydberg states. For all these three cases we divide the time into $12$ equal time steps.

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  Figure 4

  Evolution of the spin vector $⟨\mathbf{F}⟩$ for the auxiliary ($A$) and Rydberg ($R$) manifolds resulting from rf-driven Larmor precession with time-varying phases. Optimal control is based on Hamiltonian Eq.(43) for the piece-wise constant phases and total time $T_{\mathrm{tot}}=\sqrt{2}\pi /{\mathrm{\Omega }}_{\mathrm{rf}}$. The blue and black dots correspond to the first and second steps, respectively (see text). An $X=\exp (-i\pi F_x)$ gate acts on the auxiliary manifold and transfers the population from ${\overline{1}}_A$ to ${\overline{0}}_A$ and vice versa. However, for the Rydberg manifold, the pulse sequence acts as an identity operator, and the population in the ${\overline{0}}_R$ and ${\overline{1}}_R$ subspaces remain unaffected.

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  Figure 5

  Circuit diagram implementing ${\mathcal{M}}_X$. Consider an initial state $\alpha {|+⟩}_k+\beta {|-⟩}_k$, where $0\le k\le \lfloor (2J-1)/2\rfloor$. The action of the cnot gate for an ancilla state ${|+⟩}_0\equiv |+⟩$ gives us the state, $\alpha {|+⟩}_k|+⟩+\beta {|-⟩}_k|-⟩$, thus to identify whether the state is in ${|+⟩}_k$ or ${|-⟩}_k$, we need to measure whether the ancilla is in ${|+⟩}_0$ or ${|-⟩}_0$. One can achieve this using a destructive measurement [for more details see Eq.(47)].

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  Figure 6

  Circuit for error correction of a phase error for a qubit encoded in three spins. The error correction is achieved by measuring the syndromes $\{X_1{X}_2,X_2{X}_3\}$ followed by $Z=\exp (-i\pi J_z)$ gate(s) according to the syndrome outcomes.

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  Figure 7

  (a) General circuit for swapping the state of the two qubits in two different kitten subspaces. (b) The circuit that swaps the information between the data and ancilla, when the initial state of the ancilla state is ${|+⟩}_0$.

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  Figure 8

  The error-corrected logical cnot gadget. The logical $\mathrm{CNOT}$ gate is implemented by applying a physical cnot gate between each qubit (encoded in the spin) of the control and target blocks transversely. Error-correction steps are performed before and after the logical cnot.

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  Figure 9

  Imperfect amplitude error-correction gadget. There are two sources of imperfection one can associate with the amplitude error correction. The first one is a rotation error or optical pumping error occurring during the swapping approach to correct amplitude errors. The second one is due to imperfect preparation of the ancilla state, where ideally ${\rho }_A={|+⟩}_0$, however, in a nonideal setting the ancilla can be in a mixture of ${|\pm ⟩}_i$ states where $i=\{0,1,2,3,4\}$, due to optical pumping or rotation error during the state preparation. For an ideal amplitude error correction, the final state lives in the ${\mathrm{\Pi }}_0=|+⟩{⟨+|}_0+{|-⟩}_0{⟨-|}_0$, whereas for a nonideal setting, there is a small probability to be in the other manifold ${\mathrm{\Pi }}_l$. The figure shows when the final state is in the ${\mathrm{\Pi }}_i$ where $i=\{0,1,2,3,4\}$.

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  Figure 10

  Logical error as a function of the physical level error (for details of the relation between phase error and amplitude error, see Appendix pp1) for the random rotation error for a different value of $n$. Also, the threshold one needs to achieve CSS encoding in the second layer of concatenation is given for reference. Figure (a) is for the case of $p_i=0$ for $i\ne 0$ and figure (b) is for an imperfect ancilla state preparation with $p_i={10}^{-4}$ for $i\ne 0$. We can see whether the swapping error ideal or nonideal does not affect much except for very low noise and this in turn is because the contribution of the amplitude error is very low for the random rotation error. The black circle shows the intersection of the logical error with $y=x$ line for the optimal case shown here and the gray circle shows the intersection of the ${ϵ}_{\mathrm{CSS}}$ with the logical error for the optimal case. The simulation is shown for $r_1=7$ and $r_2=1$.

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  Figure 11

  Logical error as a function of the physical level error (for details to the relation between phase error and amplitude error, see Appendix pp2) for the optical pumping error for different value of $n$. The targeted threshold for the CSS code in the second layer of concatenation is given for reference. Figure (a) is for the case of $p_i=0$ for $i\ne 0$ and figure (b) is for an imperfect ancilla state preparation with $p_i={10}^{-4}$ for $i\ne 0$. We can see a significant change in the behavior depending on whether the amplitude error correction is ideal or not specifically in the low noise regime. This in turn is due to the fact that for the case of optical pumping, as seen in Appendix pp2, there is a significant contribution to the logical error from the amplitude errors. The black circle shows the threshold value for the optimal value of $n$ and the gray circle shows the intersection of the ${ϵ}_{\mathrm{CSS}}$ with the logical error for the optimal value of $n$. The simulation are shown for $r_1=7$ and $r_2=1$.

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  Figure 12

  Logical amplitude error probabilities due to rotation errors. (a) The ratio of logical amplitude error to phase error is given as a function of phase error. The probability of logical errors falls as the overall error rate decreases. A logical error occurs when we have $\lfloor (2J-1)/2\rfloor$ amplitude errors and thus as spin $J$, increases, the ratio decreases exponentially. However, for $J=3/2$, a single amplitude jump creates a logical error and thus the ratio of logical error to phase error is a constant equal to $1/2J$. (b) The ratio of logical error probability due to amplitude errors to phase error for rotation error as a function of spin $J$. We can see that this ratio exhibits an exponential trend, and the logical error becomes negligible for sufficiently large values of $J$. Consequently, fewer rounds of amplitude error correction will be needed as $J$ increases.

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  Figure 13

  The error process corresponding to the photon scattering and optical pumping for encoding a qudit in an atomic spin $\mathbf{F}$. The information is stored in the ground state and is controlled by laser light with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{L}}$ and detuning ${\mathrm{\Delta }}_{\mathrm{L}}$ from an excited-state manifold, with spin ${\mathbf{F}}^{\prime }$. Absorption of a laser photon (here $\pi$ polarized) is followed by a spontaneous emission given by wavy lines. The process causes amplitude errors and can collapse a cat state to a single magnetic sublevel.

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  Figure 14

  Evolutions of the spin vector $⟨\overrightarrow{F}⟩$ for the auxiliary ($A$) and Rydberg ($R$) manifolds resulting from rf-driven Larmor precession with time-varying phases in Eq.(43) for piecewise constant function with three time steps with a total time $T_{\mathrm{tot}}=3\pi /{\mathrm{\Omega }}_{\mathrm{rf}}$ and ${\omega }_0=5{\mathrm{\Omega }}_{\mathrm{rf}}$. For the specific choice of parameters, an $X$ gate acts on the auxiliary manifold and transfers the population from ${\overline{0}}_A$ subspace to ${\overline{1}}_A$ subspace and vice versa. However, for the Rydberg manifold, the pulse sequence acts as an identity operator, and the population in the ${\overline{0}}_R$ and ${\overline{1}}_R$ subspaces remain unaffected.

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  Figure 15

  The phase $\varphi (t)$ generates an $R=\exp (i\pi J_z)$ for the auxiliary manifold and an identity in the Rydberg manifold, which can be used to implement the rank-preserving $\mathrm{CZ}$ gate. The total time is ${\mathrm{\Omega }}_{\mathrm{rf}}T=\pi$, which is divided into ten equal time steps with ${\omega }_0=3{\mathrm{\Omega }}_{\mathrm{rf}}$ and pulse sequence is found using the quantum optimal control algorithm GRAPE.

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  Figure 16

  Circuit implementing a fault-tolerant Hadamard gate using the physical level gates for the spin-cat encoding. This differs from the standard implementation as we use both cnot and CZ gate to implement the action of the target unitary of interest.

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  Figure 17

  Circuits implementing logical level gates in ${\mathcal{C}}_1$ using the physical-level gates. (a) Logical $\mathrm{CNOT}{\mathrm{CNOT}}_{\mathrm{L}}$. To implement ${\mathrm{CNOT}}_{\mathrm{L}}$, we apply physical $\mathrm{CNOT}$ gates transversally on all qubit pairs. (b) Preparation of ${|0⟩}_{\mathrm{L}}{\mathcal{P}}_{{|0⟩}_{\mathrm{L}}}$. ${|0⟩}_{\mathrm{L}}$ is prepared by initializing the system with the state ${\mathcal{P}}_{{|+⟩}_{\mathrm{L}}}$ and measuring the parity. To measure the parity we use an ancilla initialized with ${\mathcal{P}}_{|0⟩}$ and use physical cnot gates followed by measuring the ${\mathcal{M}}_Z$, the final state is ${|0⟩}_{\mathrm{L}}$ or ${|1⟩}_{\mathrm{L}}$ for the measurement outcomes $1$ and $-1$, respectively. (c) The logical $Z$ measurement ${\mathcal{M}}_{Z_{\mathrm{L}}}$. An ancilla state is prepared in $|+⟩$ and physical CZ gates with the data qubits are applied followed by measuring the ancilla in the $X$ basis. (d) Logical $X$ measurement ${\mathcal{M}}_{X_{\mathrm{L}}}$. The logical $X$ is measured by applying the physical cnot gates and then measurement along $X$.

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  Figure 18

  Protocol for a rank-preserving Toffoli gate for spin-cat encoding using $\mathrm{SU}(2)$ operations. Similar to the rank-preserving cnot gate Fig.2, we implement the Toffoli gate in the ground state of ${}^{87}\mathrm{Sr}$ and the physical setting is the same as given in Fig.2. We consider a geometry of atoms such that the nearest neighbors are constrained by the Rydberg blockade, but the next-nearest neighbors are not constrained. In step I the population is promoted to the auxiliary manifold in the atoms. In the control atoms we promote only the population of the $\overline{0}$ subspace whereas for the target atom, the population from both the $\overline{0}$ and $\overline{1}$ subspaces are promoted to the auxiliary state. In step II, we transfer the population between the auxiliary and the Rydberg manifolds of the control atoms. In step III, we transfer the population from the auxiliary to the Rydberg manifold of the target atom. However, due to the Rydberg blockade, this population transfer only happens when both the control atoms are in $\overline{0}$ subspaces. If even one of the control atoms is in $\overline{1}$ subspace this transition is blockaded. Then similar to the rank-preserving cnot gate, in step IV we implement a $X=\exp (-i\pi J_x)$ gate in the auxiliary manifold and an identity operator in the Rydberg manifold. Finally, we will transfer all the states back to the ground state by acting steps III–I in reverse, thus implementing a rank-preserving Toffoli gate for the spin-cat encoding.

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