Noise Transfer Approach to GKP Quantum Circuits

A useful technique for analysing quantum optical experiments based on Gaussian operations and measurements is the quantum noise transfer approach BAC19 which is characterised by the displacement vector and the covariance matrix WEE12 . Working in the Heisenberg picture, the signals (the displacements) and the quantum noise (the covariances) can be tracked separately, in much the same way that signals and noise are tracked, for example, in classical communication systems. A key step in such an analysis is the definition of quadrature fluctuation operators via:

$$\delta\hat{q}=\hat{q}-q,\;\;\;\delta\hat{p}=\hat{p}-p,$$

(1)

where $q=\langle\hat{q}\rangle$ and $p=\langle\hat{p}\rangle$ are real numbers representing the average values of the position and momentum quadrature displacements respectively. Also, $\hat{q}$ and $\hat{p}$ are the position and momentum operators for the mode in question, respectively. We use the convention that the annihilation operator for the mode is given by $\hat{a}=1/2(\hat{q}+i\hat{p})$ which can then be expanded as:

$$\hat{a}=1/2(q+\delta\hat{q}+i(p+\delta\hat{p})),$$

(2)

and the classical signals (displacements) and the quantum noise operators can be independently tracked in the Heisenberg picture through various interactions and measurements in an intuitive way. In an experimental setting the noise properties of the input states can be measured directly by subtracting off the known signal and evaluating the variance of the quantum fluctuations. For example the position quadrature variance is:

$$V_{q}\equiv\langle\delta\hat{q}^{2}\rangle=\langle(\hat{q}-q)^{2}\rangle=\langle\hat{q}^{2}\rangle-q^{2}$$

(3)

The aim of this paper is to develop an analogous approach that can describe the evolution of non-Gaussian states, such as cat states RAL03 or Gottesman-Kitaev-Preskill (GKP) states GOT01 , in a similar way. Such states are formed from a superposition of differently displaced states and hence the signal is multi-valued.

In the next section we will introduce our new decomposition into signal and fluctuation operators and show how noise properties can be analysed independently of the signal value provided certain conditions are met. We illustrate this with cat state and GKP state examples. In Section 3 we consider the Heisenberg Picture evolution of our operators with loss and feedforward as examples. In Section 4 we apply our formalism to GKP states and analyse a teleportation channel which includes the GKP error correction protocol. The goal is to evaluate the noise transfer properties of the circuits based on the first and second moments of the resource states (and the properties of the feedforward measurements) – thus providing an alternative method for analysing quantum computing circuits based on GKP qubits. Current approaches to GKP circuit evaluation are mostly based on Schrödinger picture analysis. Noise can be added to ideal GKP states MEN14 and the logical states tracked through a modular sub-system decomposition PBM20 which can be generalised to finite energy GKP states TZI20 ; BOU21 . Exact numerical models incorporating noisy elements have also been explored HIL22 . In contrast the method presented here differs by describing a way to empirically quantify the noise and signals independently and then track them through circuits via their Heisenberg evolution, offering potential advantages in the intuitive nature of the approach and the tractability of the calculations in the presence of multiple noise sources. We discuss our results and conclude in Section 5.

We start by defining the operators $\delta\hat{q}$ and $\delta\hat{p}$ in an analogous way to Eq.1 via:

$$\delta\hat{q}=\hat{q}-\hat{q}_{c},\;\;\;\delta\hat{p}=\hat{p}-\hat{p}_{c}.$$

(4)

Because the displacements are multi-valued they must be represented by operators, specifically $\hat{q}_{c}$ and $\hat{p}_{c}$. We characterise $\hat{q}_{c}$ and $\hat{p}_{c}$ by their moments over restricted domains around their expected displacement values in the position and momentum quadratures respectively. Specifically, given $q_{n}=\langle\hat{q}\rangle_{n}$ is the expectation value for $q$ values falling in the $n$th position domain, we require that $\langle\hat{q}_{c}\rangle_{n}=q_{n}$. We further require that $\langle\hat{q}_{c}^{2}\rangle_{n}=q_{n}^{2}$, meaning that $\hat{q}_{c}$ has zero variance in each domain. This implies $\langle\hat{q}\hat{q}_{c}\rangle_{n}=q_{n}^{2}$. Similarly we require $\langle\hat{p}_{c}\rangle_{n}=p_{n}$ where $p_{n}=\langle\hat{p}\rangle_{n}$ is the expectation value for $p$ values falling in the $n$th momentum domain, $\langle\hat{p}_{c}^{2}\rangle_{n}=p_{n}^{2}$ and hence $\langle\hat{p}\hat{p}_{c}\rangle_{n}=p_{n}^{2}$. Assuming we have $N_{q}$ position domains and $N_{p}$ momentum domains, the full expectation values are given by

	$\displaystyle\langle\hat{q}_{c}\rangle$	$\displaystyle=$	$\displaystyle\sum_{n}^{N_{q}}q_{n}P_{nq},$		(5)
	$\displaystyle\langle\hat{p}_{c}\rangle$	$\displaystyle=$	$\displaystyle\sum_{n}^{N_{p}}p_{n}P_{np}$

with $P_{nq}$ the probability of finding a $q$ value in the $n$th domain and $P_{np}$ the probability of finding a $p$ value in the $n$th domain.

Our mode operators are now of the form:

$$\hat{a}=1/2(\hat{q}_{c}+\delta\hat{q}+i(\hat{p}_{c}+\delta\hat{p})),$$

(6)

The noise properties of the input states can be measured directly by subtracting off the closest expected displacement, evaluating the variance of the quantum fluctuations around this expected value, and taking the weighted average over all expected values. These quantities can then be related to $\delta\hat{q}$ and $\delta\hat{p}$ in the following way. We can write the average position quadrature variance described above as:

	$\displaystyle V_{q}$	$\displaystyle\equiv$	$\displaystyle\sum_{n}^{N_{q}}(\langle\hat{q}^{2}\rangle-q_{n}^{2})P_{nq}$
		$\displaystyle=$	$\displaystyle\langle\hat{q}^{2}\rangle-\sum_{n}^{N_{q}}q_{n}^{2}P_{nq}$
		$\displaystyle=$	$\displaystyle\langle\hat{q}^{2}\rangle-\sum_{n}^{N_{q}}(2\langle\hat{q}\hat{q}_{c}\rangle_{n}-\langle\hat{q}_{c}^{2}\rangle_{n})P_{nq}$
		$\displaystyle=$	$\displaystyle\langle(\hat{q}-\hat{q}_{c})^{2}\rangle$
		$\displaystyle=$	$\displaystyle\langle\delta\hat{q}^{2}\rangle.$

This can be evaluated via

$\displaystyle V_{q}$

$\displaystyle=$

$\displaystyle\int dq\;q^{2}|\Psi(q)|^{2}-\sum_{n}^{N}{{(\int_{n}dq\;q|\Psi(q)|^{2})^{2}}\over{(\int_{n}dq|\Psi(q)|^{2})}}.$

(8)

where we have used:

	$\displaystyle q_{n}$	$\displaystyle=$	$\displaystyle{{\int_{n}dq\;q|\Psi(q)|^{2}}\over{\int_{n}dq|\Psi(q)|^{2}}};$		(9)
	$\displaystyle P_{nq}$	$\displaystyle=$	$\displaystyle\int_{n}dq|\Psi(q)|^{2},$

and the integral $\int_{n}dq$ is taken over the corresponding $n$th domain whilst $\Psi(q)=\langle q|\psi\rangle$ is the position wavefunction of the state $|\psi\rangle$. Similarly, the momentum quadrature variance can be written:

	$\displaystyle V_{p}$	$\displaystyle\equiv$	$\displaystyle\langle\delta\hat{p}^{2}\rangle=\langle(\hat{p}-\hat{p}_{c})^{2}\rangle$
		$\displaystyle=$	$\displaystyle\langle\hat{p}^{2}\rangle-\langle\hat{p}_{c}^{2}\rangle$
		$\displaystyle=$	$\displaystyle\langle\hat{p}^{2}\rangle-\sum_{n}^{N_{p}}p_{n}^{2}P_{np}.$

This can be evaluated via

$\displaystyle V_{p}$

$\displaystyle=$

$\displaystyle\int dq\;p^{2}|\Psi(p)|^{2}-\sum_{n}{{(\int_{n}dp\;p|\Psi(p)|^{2})^{2}}\over{(\int_{n}dp|\Psi(p)|^{2})}},$

(11)

where the integral $\int_{n}dp$ is taken over the corresponding $n$th domain whilst $\Psi(p)=\langle p|\psi\rangle$ is the momentum wavefunction of the state $|\psi\rangle$.

If the signal peaks are well localized in their expected domains then Eqs 8 and 11 will well represent the average spread of the signal peaks. Assuming Gaussian statistics we can then estimate the probability that a quadrature measurement will find a result in their expected domain via

$$P_{j}=Erf[{{D}\over{2\sqrt{2V_{j}}}}],$$

(12)

where $D$ is the width of the domains and $j=q,p$.

The interpretation becomes more subtle if the variances approach the width of the domains. If this happens the probability distribution within the domain will be “clipped”, and will be changed relative to its “unclipped” value. We will observe this effect in our examples.

The next two subsections will examine the examples of cat states and GKP states. ‘Cat state’ usually refers to a superposition of differently displaced Gaussian states. The case we consider here is a symmetric superposition of 2 coherent states around the origin. The GKP state is a particular superposition of multiple displaced Gaussian states that has the feature that if qubit values “0” or “1” are encoded by the displacement values in the q-quadrature then the p-quadrature encodes the “+” and “-” diagonal states respectively. This nice feature means that a logical Hadamard gate can be enacted on the encoded states by a simple phase rotation. The specific GKP state construction we use here is a multiple, weighted superposition of squeezed states, displaced (and squeezed) in the q-direction. Example $q$ quadrature probability distributions for cat and GKP states are shown in Figs 1, 2 and 3.

Recall that our mode operators can be written in the form of Eq. 6. Using operators of this kind to represent the various input modes that interact in an optical circuit we can proceed to evolve them through beam-splitters, squeezers, and other quadratic unitaries (linear in the mode operators) in the usual way, whilst keeping track of their noise and signal properties. For example, suppose our initial mode, Eq. 6, passes through loss, such that only the fraction $\eta$ is transmitted. Our output mode is

$$\hat{a}_{l}=1/2(\sqrt{\eta}(\hat{q}_{c}+\delta\hat{q}+i(\hat{p}_{c}+\delta\hat{p}))+\sqrt{1-\eta}(\hat{q}_{v}+i\hat{p}_{v})),$$

(22)

where $\hat{q}_{v}$ and $\hat{p}_{v}$ are the position and momentum quadrature operators of the vacuum mode introduced by the loss, respectively. By inspection we see that the expected signal values (and hence corresponding domain boundaries) have been scaled by the factor $\sqrt{\eta}$. On the other hand the variances around these expected values are now

	$\displaystyle V_{ql}$	$\displaystyle=$	$\displaystyle\eta\langle\delta\hat{q}^{2}\rangle+(1-\eta)\langle\delta\hat{q}_{v}^{2}\rangle$
		$\displaystyle=$	$\displaystyle\eta V_{q}+(1-\eta),$

and

	$\displaystyle V_{pl}$	$\displaystyle=$	$\displaystyle\eta\langle\delta\hat{p}^{2}\rangle+(1-\eta)\langle\delta\hat{p}_{v}^{2}\rangle$
		$\displaystyle=$	$\displaystyle\eta V_{p}+(1-\eta),$

where we have used that the vacuum noise is independent and has unit variance. Considering our cat state example from the previous section we see that there is no effect on the position quadrature variance as we still have $V_{ql}=1$. The detrimental effect of loss on position only arises from the scaling down of the expected values closer to the domain boundary at $0$. On the other hand, given that for $|\alpha|>1$, $V_{p}<<1$, the effect on the momentum quadrature is a significant broadening of the peaks for relatively small amounts of loss. This combined with the reduction in the peak separation rapidly washes out the interference fringes entirely, especially for $|\alpha|>>1$. This illustrates the well-known fragility of cat states to loss.

As well as optical unitary evolution, many quantum circuits involve quadrature measurements followed by feedforward to other modes in the circuit. In particular this can occur in teleportation scenarios arising in error correction GLA06 or cluster state MEN06 protocols, or both MEN14 . Feed forward of quadrature measurements can be represented in the usual way BAC19 by feeding forward some function of the measurement operator. Thus a typical output mode $\hat{a}_{o}$ after a teleportation type circuit might be written formally as:

$$\hat{a}_{o}=1/2(\hat{q}_{co}+\delta\hat{q}_{o}+i(\hat{p}_{co}+\delta\hat{p}_{o}))+G_{1}(\hat{q}_{1})+iG_{2}(\hat{p}_{2}),$$

(25)

where feedforward from an earlier position measurement ($\hat{q}_{1}$) and an earlier momentum measurement ($\hat{p}_{2}$), have been incorporated. In standard teleportation, the “$G$” functions would be linear functions of the measurement operators, i.e. $G_{1}(\hat{q}_{1})=g_{1}\hat{q}_{1}$ and $G_{2}(\hat{p}_{2})=g_{2}\hat{p}_{2}$. However, in a situation where we can usefully write our measurement operators as $\hat{q}_{1}=\hat{q}_{c1}+\delta\hat{q}_{1}$ and $\hat{p}_{2}=\hat{p}_{c2}+\delta\hat{p}_{2}$, we can perform error correction by shifting to the nearest “ideal” result in each domain, represented by $\hat{q}_{c1}$ or $\hat{p}_{c2}$. Hence we now have the highly non-linear feed forward functions $G_{1}(\hat{q}_{1})=g_{1}\hat{q}_{c1}$ and $G_{2}(\hat{p}_{2})=g_{2}\hat{p}_{c2}$ and our output mode is now written formally as:

$$\hat{a}_{o}=1/2(\hat{q}_{co}+\delta\hat{q}_{o}+i(\hat{p}_{co}+\delta\hat{p}_{o}))+g_{1}\hat{q}_{c1}+ig_{2}\hat{p}_{c2}.$$

(26)

To illustrate feed forward with error correction we will consider a simple GKP circuit.

Another useful feature of GKP states is that small displacements of the state, as will naturally arise in the presence of loss or thermal noise, can be corrected by a straightforward circuit. This process is often referred to as GKP error correction. In the following we will analyse such a GKP error correction circuit.

The circuit we will consider is shown in Fig.6. It is an example of a continuous variable CZ teleportation circuit with error correction, where we assume both resource states are approximate GKP states and we neglect loss. We will label the approximate GKP input and two resource states with the subscripts “$1,2,3$” respectively. The resource states in modes “2” and “3” are prepared in the logical “+” state, whilst the input in mode “1” is in an arbitrary GKP logical state.

In this paper we have presented signal and noise analysis of both cat states and GKP states to highlight the generality of our approach. In particular the analysis of the preceeding section demonstrates the power and relative simplicity of this approach while also highlighting the necessity for the total noise variances to be small. Given we are grouping noise from several sources together and associating it with arbitrary logical states, it is important that we are in a regime where the noise affects different states in the same way. Fig.5 shows that this is true for GKP states provided the squeezing is of order 10 dB or higher, as was observed previously. Whilst this may seem a bit restrictive, it should be remembered that for large quantum circuits to perform faithfully they inevitably need to operate in this high fidelity regime anyway MEN14 ; BAR19 . This also applies to quantum communication applications SCH24 . Hence, we expect our approach to prove useful in tracking the flow of noise, signal and errors in quantum computing circuits employing GKP and other Bosonic qubits. Finally, we have demonstrated the power these techniques can have to develop new schemes, with a loss tolerant generalisation of the simple teleportation circuit considered allowing for general loss rates for each component.

TCR acknowledges useful discussions with Eli Bourassa, Ilan Tzitrin, Rafael Alexander and Joshua Guanzon. This work was partially supported by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No.CE110001027). SNS was supported by the Sydney Quantum Academy, Sydney, NSW, Australia.