Non-commuting two-local Hamiltonians for quantum error suppression

Abstract

Physical constraints make it challenging to implement and control many-body interactions. For this reason, designing quantum information processes with Hamiltonians consisting of only one- and two-local terms is a worthwhile challenge. Enabling error suppression with two-local Hamiltonians is particularly challenging. A no-go theorem of Marvian and Lidar (Phys Rev Lett 113(26):260504, 2014) demonstrates that, even allowing particles with high Hilbert space dimension, it is impossible to protect quantum information from single-site errors by encoding in the ground subspace of any Hamiltonian containing only commuting two-local terms. Here, we get around this no-go result by encoding in the ground subspace of a Hamiltonian consisting of non-commuting two-local terms arising from the gauge operators of a subsystem code. Specifically, we show how to protect stored quantum information against single-qubit errors using a Hamiltonian consisting of sums of the gauge generators from Bacon–Shor codes (Bacon in Phys Rev A 73(1):012340, 2006) and generalized-Bacon–Shor code (Bravyi in Phys Rev A 83(1):012320, 2011). Our results imply that non-commuting two-local Hamiltonians have more error-suppressing power than commuting two-local Hamiltonians. While far from providing full fault tolerance, this approach improves the robustness achievable in near-term implementable quantum storage and adiabatic quantum computations, reducing the number of higher-order terms required to encode commonly used adiabatic Hamiltonians such as the Ising Hamiltonians common in adiabatic quantum optimization and quantum annealing.

Appendices

Appendix 1: Rewriting the Hamiltonian in terms of auxiliary operators and stabilizers

A critical step in calculating the energy separation for a two-local error-suppressing Hamiltonian arising from subsystem codes is rewriting the Hamiltonian in terms of auxiliary operators and stabilizers. This procedure makes explicit the dependence of the error-suppressing Hamiltonian on the values of the stabilizers. It also makes easier the analytical and numerical calculations of the energy separation by reducing the size of the Hilbert space that needs to be considered and removes the degeneracy in the error-suppressing Hamiltonian. Another application of such a procedure enables exact diagonalization of a $6\times 6$ lattice of the quantum compass model [20].

Here, we describe a systematic method for finding the $m-k\,X$-type stabilizers, $m-k\,Z$-type stabilizers, and $n - 2(m-k) - k\,X$-type auxiliary logical operators, and $n - 2(m-k) - k\,Z$-type auxiliary logical operators, defining $n - 2(m-k) - k$ auxiliary qubits for a [[n, k, d]] generalized-Bacon–Shor code defined by a $m\times m$ binary matrix M of rank k. We illustrate the application of this algorithm by using it to obtain a set of auxiliary operators and stabilizers for the [[16, 2, 3]] generalized-Bacon–Shor code.

We first present pseudocode for the algorithm and then comment on its workings.

Switching the order of the rows of a matrix defining a generalized-Bacon–Shor code does not change the gauge group, only the gauge generators. Moving rows added to ${{\mathcal {R}}}_{cur}$ to the top of the matrix ensures that the auxiliary X-type operators we define commute with all previously defined auxiliary operators. Specifically, the Z-type operators previously defined do not involve this row. The Z-type operators we define are all within ${{\mathcal {R}}}_{cur}$, because the constraint of minimal linear dependence guarantees that there will be a nonzero entry in the column below each qubit in the top row. We exclude the left-most column Z-type operator because it is the product of the stabilizer and the other Z-type operators which we define. Because we will not be including that column, we choose X-type operators that include the first qubit since that enables us to satisfy easily the canonical commutation relation for X- and Z-type operators defining auxiliary qubits. Once we are considering columns, we must be more careful. While the Z-type operators we define automatically commute with all previously defined operators, because those rows were not used to extract X-type operators, this property is not guaranteed for the X-type operators, which is why the extraction of X-type operators at this state is more complicated than for Z-type operators.

1.1 Example: Extracting auxiliary operators and stabilizers for the [[16,2,3]] code

The following symmetric matrix defines a generalized-Bacon–Shor [[16, 2, 3]] code,

(25)

The subscripts in $M_{5\vert 5}$ indicate the numbers of the rows and columns, respectively. We use $c_j$ and $r_k$ to label the row and column indices, so that we can keep track of the qubits after row and column manipulations. The rank of the matrix $M_{5\vert 5}$ is 2, and the distance of the code is 3 (since any linear combination of its rows or columns has either at least three 1 s or none). We will use the algorithm to find the 3 X-type stabilizers, the 3 Z-type stabilizers, and the 8 auxiliary qubits defined by 8 pairs of auxiliary operators satisfying the canonical commutation relations.

Moving row $r_3$ to the top of the matrix yields

(26)

where we underline the row labels to indicate the rows (columns) we are currently considering, rows that are in ${{\mathcal {R}}}_{cur}$. Since the top two rows in $M_{5\vert 5}'$ are identical, and thus linearly dependent, we define a stabilizer $S^Z_1 = R^Z_3 R^Z_1$, where $R^Z_3 = Z_{3,2}Z_{3,4}Z_{3,5}$ and $R^Z_1 = Z_{1,2}Z_{1,4}Z_{1,5}$. We now extract auxiliary operators as we eliminate the top row. We define two auxiliary operators $X_{A1} = X_{3,2}X_{3,4}$ and $X_{A2} = X_{3,2}X_{3,5}$, and define the corresponding Z-type auxiliary operators to be $Z_{A1} = Z_{3,4}Z_{1,4}$ and $Z_{A2} = Z_{3,5}Z_{1,5}$. It is easy to check that two pairs of auxiliary operators satisfy the standard commutation relations. Note $Z_{3,2}Z_{1,2} = S^Z_1 Z_{A1} Z_{A2}$. Having used the top row to obtain auxiliary operators and stabilizers, we may remove the top row. We consider the resulting matrix

(27)

A minimally linearly independent set on which the top row is linearly dependent is $\{r_2, r_5\}$. We move these rows to the top of the matrix to obtain

(28)

We define a stabilizer $S^Z_2 = R^Z_2 R^Z_5 R^Z_1$, where $R^Z_2 = Z_{2,1}Z_{2,3}Z_{2,5}, R^Z_5 = Z_{5,1}Z_{5,2}Z_{5,3}Z_{5,4}$, and $R^Z_1 =Z_{1,2}Z_{1,4}Z_{1,5}$. We define X-type auxiliary operators $X_{A3} = X_{2,1}X_{2,3}$ and $X_{A4} = X_{2,1}X_{2,5}$, and Z-type auxiliary operators $Z_{A3} = Z_{2,3}Z_{5,3}$, and $Z_{A4} = Z_{2,5}Z_{1,5}$. We may now remove the top row to obtain

(29)

We move $r_4$ to the top of the matrix to obtain

(30)

We define a stabilizer $S^Z_3 = R^Z_4 R^Z_5 R^Z_1$, where $R^Z_4 =Z_{4,1}Z_{4,3}Z_{4,5}, R^Z_5 = Z_{5,1}Z_{5,2}Z_{5,3}Z_{5,4}$, and $R^Z_1 = Z_{1,2}Z_{1,4}Z_{1,5}$. We define X-type auxiliary operators $X_{A5} = X_{4,1}X_{4,3}$ and $X_{A6} = X_{4,1}X_{4,5}$, and Z-type operators $Z_{A5} = Z_{4,3}Z_{5,3}$ and $Z_{A6} = Z_{4,5}Z_{1,5}$. We may now remove the top row to obtain

(31)

Now that the rows are linearly independent, we can engage “column elimination” to extract further operators. We move $c_3$ to the far left to obtain

(32)

Were $M'_{2|5}$ the starting matrix, the first X-type stabilizer would be ${\tilde{S}}^X_1 = X_{5,3}X_{5,1}$. However, this X-type stabilizer does not commute with some of the Z-type auxiliary operators we introduced before. To obtain the correct stabilizer, we need to iteratively multiply ${\tilde{S}}^X_1$ by X-type operators corresponding to (anti-commuting with) Z-type operators that do not commute with ${\tilde{S}}^X_1$. It does not commute with $Z_{A5} = Z_{4,3}Z_{5,3}$, so we need to multiply by $X_{A5} = X_{4,1}X_{4,3}$, which in turn does not commute with $Z_{A3} = Z_{2,3}Z_{5,3}$, so we need to multiply by $X_{A3} = X_{2,1}X_{2,3}$. The result is the stabilizer

$$\begin{aligned} \begin{aligned} S^X_1&= X_{2,3}X_{4,3} X_{5,3} X_{2,1} X_{4,1} X_{5,1} \\&= C^X_3 C^X_1. \end{aligned} \end{aligned}$$

(33)

The left-most column contains no pairs of 1s, so we do not extract any auxiliary operators at this step. We may now remove the far left column to obtain

(34)

Columns $c_2$ and $c_5$ form a minimally linearly independent set on which the left-most column $c_1$ depends. We therefore move these columns to the far left to obtain

(35)

A similar argument to the one above leads to defining the stabilizer

$$\begin{aligned} \begin{aligned} S^X_2&= X_{1,2} X_{3,2} X_{5,2} X_{1,5} X_{2,5} X_{3,5} X_{4,5} X_{2,1} X_{4,1} X_{5,1}\\&= C^X_2 C^X_5 C^X_1 . \end{aligned} \end{aligned}$$

(36)

We also define the pair of operators $Z_{A7}=Z_{5,2}Z_{1,2}$ and ${\tilde{X}}_{A7}=X_{1,2}X_{1,5}$. Since ${\tilde{X}}_{A7}$ anti-commutes with $Z_{A2}, Z_{A4}$, and $Z_{A6}$, we have $X_{A7}={\tilde{X}}_{A7}X_{A2}X_{A4}X_{A6}$. We now remove the left-most column to obtain

(37)

Moving the row $c_4$ to far left, we have

(38)

These columns contribute a stabilizer

$$\begin{aligned} \begin{aligned} S^X_3&= C^X_4 C^X_5 C^X_1 \\&= X_{1,4} X_{3,4}X_{5,4} X_{1,5}X_{2,5}X_{3,5}X_{4,5} X_{2,1}X_{4,1} X_{5,1}. \end{aligned} \end{aligned}$$

(39)

We also extract the final set of auxiliary operators $Z_{A8} = Z_{5,4}Z_{1,4}$ and $X_{A8} = {\tilde{X}}_{A8} X_{A1} X_{A2} X_{A4}X_{A6}$, where ${\tilde{X}}_{A8} = X_{1,4}X_{1,5}$. Removing the first column results in a matrix with linearly independent rows and columns and in which no row or column contains a pair of 1s; therefore, we stop extracting operators. Indeed, we have already obtained 3 X-type and 3 Z-type stabilizers and 8 pairs of auxiliary operators as expected.

In conclusion, we have following auxiliary operators

$$\begin{aligned} X_{A1}= & {} X_{3,2}X_{3,4},\quad X_{A2} = X_{3,2}X_{3,5},\quad X_{A3} = X_{2,1}X_{2,3},\nonumber \\ X_{A4}= & {} X_{2,1}X_{2,5},\quad X_{A5} = X_{4,1}X_{4,3},\quad Z_{A1} = Z_{1,4}Z_{3,4},\nonumber \\ Z_{A2}= & {} Z_{1,5}Z_{3,5},\quad Z_{A3} = Z_{2,3}Z_{5,3},\quad Z_{A4} = Z_{2,5}Z_{1,5},\nonumber \\ Z_{A5}= & {} Z_{4,3}Z_{5,3},\,\quad X_{A6} = X_{4,1}X_{4,5},\quad X_{A7}=X_{1,2}X_{1,5}X_{A2}X_{A4}X_{A6},\nonumber \\ X_{A8}= & {} X_{1,4}X_{1,5}X_{A1} X_{A2} X_{A4} X_{A6} ,\quad Z_{A6} = Z_{4,5}Z_{1,5},\nonumber \\ Z_{A7}= & {} Z_{5,2}Z_{1,2},\quad Z_{A8} = Z_{5,4}Z_{1,4}, \end{aligned}$$

(40)

and the stabilizers take the form

$$\begin{aligned} S^X_1= & {} C^X_3C^X_1,\quad S^X_2 = C^X_2C^X_5C^X_1,\quad S^X_3 = C^X_5 C^X_1 C^X_4,\nonumber \\ S^Z_1= & {} R^Z_1 R^Z_3,\quad S^Z_2 = R^Z_2R^Z_5 R^Z_1,\quad S^Z_3 = R^Z_5 R^Z_1 R^Z_4\,. \end{aligned}$$

(41)

Appendix 2: Noise model

For our numerical analyses of error suppression, we consider the spin-boson Hamiltonian [18],

$$\begin{aligned} H(t) = H_S(t) + H_{B} + \sum _{k=1}^n \Big (X_k\otimes B_k^X+ Y_k\otimes B_k^Y + Z_k\otimes B_k^Z\Big ), \end{aligned}$$

(42)

where $H_S = H_\mathrm {supp}+ H_L$ is the system Hamiltonian, and $X_k, Y_k$, and $Z_k$ are Pauli operators acting on the kth qubit. The sum in Eq. (42) describes interactions between individual Pauli operators of the system qubits and independent bath modes, where

$$\begin{aligned} B_k^X= & {} \sum _\mu g_\mu ^X\big ( a_{\mu , k} + a_{\mu , k}^\dagger \big ),\end{aligned}$$

(43)

$$\begin{aligned} B_k^Y= & {} \sum _{\nu } g_\nu ^Y \big (b_{\nu , k} + b_{\nu , k}^\dagger \big ),\end{aligned}$$

(44)

$$\begin{aligned} B_k^Z= & {} \sum _{\tau } g_\tau ^Z \big (c_{\tau , k} + c_{\tau , k}^\dagger \big ), \end{aligned}$$

(45)

with $g_\mu ^X, g_\nu ^Y$, and $g_\tau ^Z$ being the coupling constants. We consider the case in which all of these coupling constants have the same value. The term $H_{B}$ in Eq. (42) is the bath Hamiltonian,

$$\begin{aligned} H_{B} = \sum _{\mu , j} \hbar \, \omega _{\mu , j}^X\, a^\dagger _{\mu , j} a_{\mu , j} + \sum _{\nu , k} \hbar \, \omega _{\nu , k}^Y\, b^\dagger _{\nu , k} b_{\nu , k} + \sum _{\tau , l}\hbar \, \omega _{\tau , l}^Z\, c^\dagger _{\tau , l} c_{\tau , l}. \end{aligned}$$

(46)

Going to the Heisenberg picture of the bath Hamiltonian, we have

$$\begin{aligned} B_k(t) = e^{itH_{B}/\hbar } B_k\, e^{-itH_{B}/\hbar } = \sum _\mu \Big ( g_\mu a_{\mu , k} e^{-i\omega _\mu t} + g_\mu ^*a_{\mu , k}^\dagger e^{i\omega _\mu t}\Big ), \end{aligned}$$

(47)

where the superscripts X, Y, and Z are neglected for abbreviation of notation. The bath correlation function then takes the form

$$\begin{aligned} \begin{aligned} C_\mathrm {bath}(j,t;\,k,t')&=\big \langle B_j(t) B_k(t')\big \rangle \\&= \delta _{j,k}\sum _\mu \, \vert g_\mu \vert ^2 \Big ( \big \langle a_{\mu , k}^\dagger a_{\mu , k}\big \rangle \, e^{-i\omega _{\mu , k} (t-t')} + \mathrm {c.c.}\Big )\\&= \delta _{j,k} C_\mathrm {bath}(t-t'). \end{aligned} \end{aligned}$$

(48)

The expectation values in Eq. (48) satisfy the Planck condition for thermal baths,

$$\begin{aligned}&\big \langle a_{\mu }^\dagger a_{\mu }\big \rangle = \frac{1}{e^{\hbar \omega _\mu /k_\mathrm{B} T}-1}, \quad \; \big \langle a_{\mu } a_{\mu }^\dagger \big \rangle = \big \langle a_{\mu }^\dagger a_{\mu }\big \rangle + 1 = \frac{1}{1-e^{-\hbar \omega _\mu /k_\mathrm{B} T}}, \end{aligned}$$

(49)

where the qubit subscript k is omitted. The Fourier transformation of the bath correlation function is

$$\begin{aligned} {\widetilde{C}}_\mathrm {bath}(\omega ) = \int dt\, e^{-i\omega t}C_\mathrm {bath}(t)= \frac{2 \pi J(\vert \omega \vert )}{\left| 1- e^{-\hbar \omega /k_\mathrm{B} T}\right| }, \end{aligned}$$

(50)

where $J(\omega )$ is the bath spectral function arising from the substitution of the sum in Eq. (48) with an integral,

$$\begin{aligned} \sum _\mu \vert g_\mu \vert ^2 \simeq \int _0^\infty d\omega \, J(\omega ). \end{aligned}$$

(51)

The bath correlation function determines the transition rate from one system state $|{\psi _\alpha }\rangle$ to another state $|{\psi _\beta }\rangle$, where $E_\alpha$ and $E_\beta$ are the energies of the two states and $Q$ is the system noise operator (in this case X, Y, and Z). The ratio of the transition rates between any two states satisfies

$$\begin{aligned} \frac{\varGamma _{\psi _\alpha \rightarrow \psi _\beta }}{ \varGamma _{\psi _\beta \rightarrow \psi _\alpha }} = e^{(E_\alpha -E_\beta )/k_\mathrm{B} T}, \end{aligned}$$

(52)

which gives the correct population ratio of $|{\psi _\beta }\rangle \langle {\psi _\beta }|$ and $|{\psi _\alpha }\rangle \langle {\psi _\alpha }|$ at thermal equilibrium, i.e., the Boltzmann distribution. The function ${\widetilde{C}}_\mathrm {bath}(\omega )$ determines the transition rate from a lower-energy state to a higher-energy state when $\omega <0$, and the other way around when $\omega > 0$. While transitions to higher-energy states are detrimental, transitions to lower-energy states are beneficial for adiabatic quantum computation.

We further assume that the bath spectral function satisfies the Ohmic condition,

$$\begin{aligned} J (\omega ) \simeq \hbar ^2 \chi \, \omega e^{-\omega /\omega _c},\quad \mathrm {for}\;\omega \ge 0, \end{aligned}$$

(53)

where $\chi$ is a dimensionless constant and $\omega _c$ is the cutoff frequency. The bath correlation function can thus be simplified to

$$\begin{aligned} {\widetilde{C}}_\mathrm {bath}(\omega ) = \frac{2 \pi \hbar ^2 \chi \omega e^{-\vert \omega \vert /\omega _c}}{1- e^{-\omega /\omega _T}}, \end{aligned}$$

(54)

where $\omega _T= K_\mathrm{B}T/\hbar$. We plot this function with the parameters given in [19]: $\chi = 3.18 \times 10^{-4}, \omega _c=8\pi \times 10^9\, \mathrm {rad/s}$, and $\omega _T= 2.2\times 10^9\, \mathrm {rad/s}$ (at $17\,\mathrm {mK}$). At zero frequency, i.e., transitions between two states of the same energy, ${\widetilde{C}}_\mathrm {bath}(0) = 2 \pi \hbar ^2 \chi \, \omega _T$ is proportional to the temperature T. The derivative of ${\widetilde{C}}_\mathrm {bath}(\omega )$ is not continuous at $\omega = 0$ due to the finite cutoff frequency $\omega _c$. The function ${\widetilde{C}}_\mathrm {bath}(\omega )$ decays quickly once $\omega$ is smaller than $-\omega _T$; the transition rate to higher-energy states is low for an energy difference that is several times larger than $K_\mathrm{B}T$. Consequently, an energy gap as large as several times of $k_\mathrm{B} T$ can keep the system in the ground state for a much longer time than the gapless case. The asymmetry of the function ${\widetilde{C}}_\mathrm {bath}(\omega )$ (see Fig. 6) can also be used to prepare the initial state when the energy gap is less than $\hbar \omega _c$ but larger than $\hbar \omega _T$, as the noise terms drive the system to its ground state while the opposite effect is suppressed.

Fig. 6

Bath correlation function for Ohmic noise