Abstract
The ability to perform computations on encrypted data is a powerful tool for protecting a client’s privacy, especially in today’s era of cloud and distributed computing. In terms of privacy, the best solutions that classical techniques can achieve are unfortunately not unconditionally secure in the sense that they are dependent on a hacker’s computational power. Here we theoretically investigate, and experimentally demonstrate with Gaussian displacement and squeezing operations, a quantum solution that achieves the security of a user’s privacy using the practical technology of continuous variables. We demonstrate losses of up to 10 km both ways between the client and the server and show that security can still be achieved. Our approach offers a number of practical benefits (from a quantum perspective) that could one day allow the potential widespread adoption of this quantum technology in future cloudbased computing networks.
Introduction
Incredibly, 2.5 quintillion bytes of data are produced in the world every single day. In fact, it has been estimated that >90% of the world’s data was created in the past 2 years^{1}. Most of these data are stored around the world in data centres and accessed remotely via the cloud. Because cloud computing is operated by third parties (for example, Amazon, Facebook), one of the outcomes of this acceleration in information is the need to better protect our privacy. In principle, the cloud contains various types of data for which security and privacy are essential. For example, an individual’s personal data (such as medical records and credit card information), the trade secrets and intellectual property of multinational corporations and sensitive government information (for example, the Central Intelligence Agency bought cloud space from Amazon). Therefore securing a client’s privacy in the cloud is one of the core security challenges we face today.
One of the current solutions to this challenge is homomorphic encryption^{2}. The requirement for such a solution was first identified in the 1970s by Rivest et al.^{3} It was not until over 30 years later that Craig Gentry, in his Stanford PhD dissertation, discovered fully homomorphic encryption^{4}. Although there has been much progress in recent years, the best known implementations of fully homomorphic encryption are impractical for today’s computers^{2,5,6,7}.
One might hope that a generalization of homomorphic encryption to quantum computations would be less demanding. If one restricts the class of quantum operations to be implemented, then it was shown one can hide up to a constant fraction, which can be made arbitrarily close to unity, of the encrypted information while only requiring polynomial overhead^{8}. Unfortunately, it has been shown that perfectly secure, deterministic fully homomorphic quantum computation is only possible at the expense of an exponential overhead^{9}. One can relax the requirements of quantum homomorphic encryption by allowing further rounds of interaction between the client and server. Such a scheme was first studied by Childs^{10} where he outlined a protocol to allow an individual of limited quantum ability (a client) to delegate a quantum computation to another person (a server) who is in possession of a fully fledged quantum computer. The idea here being that such a computer is only initially available to a select few. Progress towards this direction was demonstrated recently by IBM who made available a small (5 qubit) quantum computer for access in the cloud^{11}. The crux of the issue in developing such a protocol lies in the fact that the client wants to hide some subset of her input, the quantum program, and the final result. That such a scheme is even possible is astounding from a classical viewpoint. Seminal work by Broadbent et al.^{12} built upon this idea, in the clusterstate framework^{13}, to develop the notion of universal blind quantum computing: a protocol that fulfils all three of the above criteria and requires only that the client is able to prepare and send single qubits from a finite set.
The field of delegated quantum computing has since exploded with vast interest in both theoretic advances and experimental demonstrations^{14,15,16,17,18,19,20,21,22}. Unfortunately, accomplishing all three of the client’s criteria places stringent requirements on the experimental realization of such protocols. One might hope that by giving up one of these demands, less resources would be required in a physical implementation, while still providing a much needed solution. Interesting work by Fisher et al.^{23} showed that if the client and server agree on the program beforehand, then one requires less quantum and classical communication. Such a protocol can be termed quantum computing on encrypted data. This scheme offers the advantage of allowing one to carry out quantum computations on encrypted data with relatively low overhead, namely, generation of random states and classical key updates.
In this paper, we offer an approach to quantum computing on encrypted data that does not require challenging single photon sources or single photon detectors and is based on a different type of substrate known as continuous variables (CVs)^{24,25}. CVs offer a number of practical advantages over its qubit counterpart: deterministic gate implementation, low cost and affordability of components (such as laser sources and detectors), high detection efficiencies (at room temperature), high rate of information transfer, and the ability to be fully integrated within current telecommunication infrastructures. Here we answer the questions of which gates the client is required to perform, how much classical communication is required between the client and server and how many classical/quantum operations are needed per gate in our protocol. Furthermore, we provide proofofprinciple experimental results that highlight the effect of loss equivalent to over 10 km of fibre transmission at a telecom wavelength.
Results
Theory
Our protocol for computing on encrypted CVs consists of three stages (cf. Fig. 1): an encryption stage, a program stage, and a decryption stage. We will now elaborate in more detail. First, the client performs an encryption operation on their desired input to limit the amount of information the server can obtain about the initial state. The state is then sent to the server, who performs a predetermined set of gates known to both parties (corresponding to the program needed to be performed). Finally, the state is sent back to the client who is able to perform a decryption operation that recovers the output of the desired computation. To discuss the encryption operation, we first define the Heisenberg–Weyl operators^{25} and , as well as the displacement operator where are the canonical amplitude and phase operators, respectively, which obey Heisenberg’s uncertainty relation . The annihilation and creation operators are denoted by respectively, and are defined by and its adjoint. Consider the action of applying a random displacement in phase space to the input state; intuitively speaking, if the displacement is chosen randomly then the state will look mixed and smeared out over all of phase space to somebody unaware of the displacement parameters. This can be made rigorous by invoking the fact that holds for any normalized state . This shows us that averaging the displacements applied to any normalized state, over the entire complex plane, results in a quantity proportional to the identity. In reality, a uniform distribution of displacements over is unphysical and we must replace this with a function that dies off sufficiently fast in order to adhere to some energy threshold. For a fixed energy, a Gaussian distribution will maximize the entropy of the resulting state^{26} and thus we restrict ourselves to Gaussian distributions. This will potentially give the server the capability of extracting some information about the input state, but the amount will be bounded based on the width of the Gaussian. A formal security analysis of our protocol, in the limit of finite squeezing and displacements, remains an open problem.
We now turn our attention to the server who is asked to perform some known algorithm on the encrypted data, where, in principle, the algorithm corresponds to a universal set of quantum gates. To show that the server can perform such a computation, it suffices to show that we can implement a universal set of gates. Namely, we need to show that the set where , and can be implemented and decrypted with an appropriately chosen operation. Note that of the gates in this set only the U_{3}(T) gate^{27,28} is not Gaussian, and we only require this one nonGaussian gate in order to achieve universal quantum computation^{29}. However, it is prudent to note that nonGaussian operations are very challenging to implement and that there remain many practical challenges in devising a robust quantum computer based on CVs^{29}. Except for U_{3}, all of these operations have decryption operators that correspond to displacements (cf. Table 1), and this allows for the straightforward composition of gates. To make it clearer, we consider a simple example, namely, the Z(T) gate. The encryption operation D(Q, P) consists of a translation in both the amplitude and phase quadratures and it can be decomposed as a sequence of an X(Q) as well as a Z(P) gate, the latter commutes with Z(T) and so will simply slide through the gate. Consider the application of the X(Q) gate: this gate slides through the Z(T) gate up to a phase as X(Q)Z(T)=e^{−iQT}Z(T)X(Q). Thus we can construct a decryption operation as
where C(Q, P, T)=exp[i(QP/2−QT)]X(−Q)Z(−P) is the decryption gate which, when applied, will undo the effect of the initial encryption operation. However, the U_{2} decryption gate present in the U_{3} operation does not easily slide through the Fourier gate F, and thus we must have the server correct for this onthefly; this is possible in a manner similar to the discretevariable protocol presented in ref. 23. To perform U_{3}(T), the client instead sends the server two modes, the first of which is the encrypted state and the second being the state U_{2}(A)Z(Q′)0〉_{p}, where A and Q′ are chosen randomly and 0〉_{p} denotes a momentum eigenstate; finite squeezing does not present any issues other than the ones normally associated with teleportation, namely, the introduction of extra Gaussian noise^{30}. The server is then able to implement U_{3}(T) as shown in the circuit in Fig. 2. After the client sends both modes and the value of B, the server performs the desired U_{3}(T) gate after application of the inverse Fourier gate, before interacting the two modes with a controlled phase gate, indicated by the vertical line. The server then measures the first mode and after performing the U_{2}(B) gate obtains the desired state U_{3}(T) on the remaining mode, up to displacement corrections, provided that B=−3QT−A. One can view this decomposition of −3QT as an additive white Gaussian noise channel where the input is modulated by a noise source A; as the variance of A increases the channel capacity approaches zero and thus the server obtains no information about the encryption parameter Q. Note that the additional corrections depend on the value m_{1}A and so the server must communicate the value of m_{1} obtained to the client, thus requiring one round of classical communication for the full implementation. In the limit of ideal teleportation and infinite variance in the Gaussian probability density functions governing the selection of A and Q′, it can be shown that this interactive protocol does not compromise the encryption; the proof follows identically to that in ref. 23. A proof in the presence of both finite squeezing and displacements remains an open challenge.
We discuss the implementation of this protocol in more detail as well as a discussion of the decryption operations for each gate in the universal set in Supplementary Note 1. We provide additional details in Supplementary Information, in particular: how to compose gates (Supplementary Note 2), the effects of transmission (Supplementary Note 3) and imperfect encryption (Supplementary Note 4), an entanglementbased analogue (Supplementary Note 5), the use of channel estimation (Supplementary Note 6), and limitations on squeezing (Supplementary Note 7).
Experiment
The CV quantum gates associated with linear phase space displacements and squeezing transformations allow for an experimental test of quantum computing on encrypted data solely based on Gaussian quantum states and Gaussian operations. Such operations can be performed with high fidelity within the field of CV quantum optics. In the following, we thus use Gaussian displacement and squeezing operations to test the basic principles of quantum computing on encrypted data, that is, we implement a server performing first Z and X gates and second a squeezing gate, related to the U_{2} gate.
We start by testing the effectiveness of the encryption operation by using the experimental setup shown in Fig. 3a. Quantum information at the location of the client was generated in the form of a coherent state of light ϕ〉=α〉. To test the protocol for many different coherent state excitations simultaneously, we produced an ensemble of coherent states by means of a set of electrooptical modulators (EOMs), thereby preparing the Gaussian ensemble where G_{in}(Q, P) is a Gaussian probability density function with variance V_{in}. This information was then encrypted by applying a randomized phase space displacement onto the coherent state ensemble using the same two EOMs driven by two independent Gaussian white noise sources with equal variances V_{Q}=V_{P}=V_{enc} for the amplitude (Q) and phase (P) quadratures. This encryption noise results in an encrypted state where G_{tot}(Q, P) is a Gaussian distribution with a total variance of V_{in}+V_{enc}. We measured the encrypted quantum states with homodyne detection and recorded the correlations between the measurements and the input signal. Using these correlations, we calculate the mutual information as plotted in Fig. 3b. The solid line is a theoretical prediction given by
For efficient encryption, the quadrature correlations and thus the mutual information between the encrypted state received by the server and the input state prepared by the client should be vanishingly small. From the plot, we clearly see the effects of a finite encryption variance.
We first implement the Z and X displacement gates as illustrated in Fig. 4a. The protocol was performed with a Gaussian alphabet of coherent states with variance V_{in}=0.28 shot noise units (SNU) embedded in encryption noise of V_{enc}=31 SNU. For this particular encryption, the mutual information is I=0.005 bits per use. The Z and X gates were tested for a symmetric Gaussian distribution of displacements with variance V_{gate}=0.6 SNU. This results in the state ρ after the computation having a total variance of V_{gate}+V_{in}+V_{enc}.
Finally, the state is sent back through a lossy channel to the client who is decrypting the state using two EOMs driven by noise that was optimally anticorrelated with the noise used for encryption. The final state is then ideally given by D(Q, P)ϕ〉, but owing to imperfections there is some residual noise from the encryption protocol and thus we must consider the output state ρ with a variance of V_{in}+V_{gate}+V_{res}. For the presented measurement, the residual noise was V_{res}=0.072 SNU. To visualize the evolution of the information content at different stages of the scheme, in Fig. 4b, we plot the signaltonoise ratios (SNRs) of a single quadrature after each stage of the protocol. It is clear from these numbers that the amount of information in the encrypted state is close to zero and that the decryption operation is almost ideal. For further quantification, we show in Fig. 4c the fidelity between the ensembles of output states of the quantum computation using encrypted states and plaintext states. The fidelities are >97% for all measured transmission values.
We now turn to the implementation of the squeezing gate, which is defined as , where r is the squeezing parameter. It is directly related to the U_{2}(T) gate by two additional phase shifts and a suitable transformation^{31} between T and r, see Supplementary Note 8 for a full justification. Figure 5a shows the experimental setup. In contrast to the implementation of the displacement gates, we used a single coherent excitation rather than ensembles of coherent input states. The squeezedlight source was based on parametric downconversion in a periodically poled potassium titanyl phosphate crystal placed in a linear cavity. In our realization of the gate, we directly squeezed the (encrypted) input state in the squeezedlight source. We note that this constitutes the first demonstration of an inline squeezing transformation of quantum information. Previous demonstrations have relied on offline squeezed states^{31}. A full Wigner function illustration is presented in Fig. 5c–f measured by homodyne detection after each of the four steps. The state after the squeezing gate is shown in Fig. 5e, and it is clear that the squeezing operation is hardly visible as a result of the encryption noise. Finally, the transformed state is decrypted through displacements at the client, thereby revealing the output state of the server which is displayed in Fig. 5f. The squeezing transformations of the first and second moments of the state are now clearly visible.
We quantify the performance of the squeezing protocol on encrypted data by computing the fidelity between the state retrieved by the client after decrypting the computed input state and the state received by the client when no encryption was used. This comparison is carried out under the variation of the amount of encryption noise for both no transmission loss and loss corresponding to 10 km fibre propagation at a telecom wavelength, equivalent to 5 dB transmission loss. This loss is implemented with a halfwave plate and polarizing beam splitter combination. The results are presented in Fig. 5b. The variation of the fidelities, as was also observed for the displacement gates, comes mainly from systematic errors in the fine tuning of the decryption and system drifts. The deviation from unity fidelity is mainly caused by imperfect reconstruction of the Wigner functions and system drifts as well as nonideal decryption owing to a small amount of decorrelation between the encryption and decryption noise. Despite these imperfections, the fidelity is close to unity and stays >98.5% for all parameters. In general, the fidelities for the loss case are a bit higher owing to the fact that the loss forces the decrypted states closer to the vacuum state. The measured high fidelities show that performing quantum computing on encrypted states rather than on plaintext states is indeed feasible.
To quantify the performance of the implemented remote gate itself, we display in Fig. 5g the SNRs of the Q and P quadratures for the different steps of the protocol. Ideally, that is, without channel loss, without loss in the gate and with perfect decryption, the squeezing gate will preserve the SNR as indicated by the two dashed lines. For the output state, we display the SNRs for both with and without encryption, showing a small decrease in SNR if encryption is used. The remaining reduction of the SNR in comparison to the ideal gate is mainly due to optical loss, that is, about 2.2 dB for the gate, including the squeezer and the supply optics. A more detailed loss analysis can be found in Supplementary Methods.
Discussion
We have developed a continuousvariable protocol for quantum computing on encrypted variables where we required a baseline of only two uses of a quantum channel: one use for the input and another for the output. We required one additional round of classical communication in each direction and one additional use of the quantum channel to implement a cubic phase gate U_{3}(T), while Gaussian gates can be implemented with no communication cost. The client needs only be capable of performing displacements to both encrypt and decrypt, except when they perform a U_{3}(T) gate the client must be capable of implementing a U_{2}(A) gate as well. Alternatively, one could run an equivalent entanglementbased version of this protocol, which relies on teleportation^{32}. To achieve highfidelity teleportation, one could use a hybrid teleportation scheme such as that of ref. 33.
In this paper, we also studied how to compose gates for encryption and decryption, the effects of transmission and imperfect encryption, an entanglementbased analogue and the use of channel estimation as well as the limitations on squeezing in the cubic phase gate. We have experimentally demonstrated our scheme in performing both displacements and online squeezing operations on an alphabet of coherent states and studied the resulting performance in terms of the fidelity. Finally, to the best of our knowledge, this is the first time quantum computing on encrypted data has been generalized to CVs as well as the first proofofprinciple demonstration of any form (qubit or qumode) of secure delegated quantum computing over a lossy channel. Extending this protocol to long distances will inevitably require quantumrepeater technologies to preserve the encrypted states^{34}. We hope our results will help lay the ground work for future theoretical explorations and experimental demonstrations similar to those for quantum key distribution, such as free space and field demonstrations.
Methods
State preparation and gate operations
In all experiments, state preparation was carried out using a coherent continuouswave 1,064 nm laser beam as a carrier. The investigated quantum states were prepared at 10.5 MHz sidebands relative to the carrier, where the spectrum was shotnoise limited. Coherent states were prepared in these sidebands by injecting the laser beam into a pair of EOMs, which were driven with 10.5 MHz sinusoidal signals. Thermal states were prepared by applying two independent white noise signals around 10.5 MHz to a pair of EOMs. Squeezed states were prepared using a semimonolithic cavity with a temperaturestabilized periodically poled potassium titanyl phosphate crystal as a nonlinear medium, pumped by 532 nm light. The cavity length was locked using the Hänsch–Couillaud locking scheme^{35}. The relative phase between the pump and the quantum state carrier beams was locked through a 36.7 MHz sideband lock applied onto the carrier beam by the EOM, which also produces the input state in the protocol. See Fig. 5.
Measurements
Output states were measured using scanned homodyne detection. The detector’s output was mixed with an electronic local oscillator at 10.5 MHz in an analogue mixer. The downmixed output was amplified by an amplifier with adjustable gain and lowpass filtered at 1 MHz. The filtered signal was digitized with a 14 bit digitaltoanalogue converter at a rate of 5 MHz.
Data processing
The digitized signal was filtered with a 10 kHz highpass filter to suppress 50 Hz fluctuations in the measurements originating from the voltage supply. The density matrices of the measured states were reconstructed from the digitized and filtered data through the use of maximum likelihood and filtered backprojection algorithms^{36}, depending on the mean photon number of the state. The fidelity was estimated from the reconstructed density matrices, using the equation^{25}
SNRs were determined from the estimated variances and mean values of the reconstructed states.
Data availability
Data and data processing scripts are available from the corresponding authors upon request.
Additional information
How to cite this article: Marshall, K. et al. Continuousvariable quantum computing on encrypted data. Nat. Commun. 7:13795 doi: 10.1038/ncomms13795 (2016).
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Acknowledgements
T.G. and U.L.A. acknowledge support from the Danish Research Council for Independent Research (Individual Postdoc and Sapere Aude, 418400338B and 10081599). We also acknowledge support from Villum Fonden project no. 13300. K.M. acknowledges support from the Natural Sciences and Engineering Research Council of Canada.
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K.M. and C.W. developed the theory. C.S.J., T.G. and U.L.A. conceived the experiment. C.S.J. and C.S. performed the experiment under the supervision of T.G. and U.L.A. and the data was analysed by C.S.J. The manuscript was prepared by K.M., C.S.J., T.G. and U.L.A. with contributions from all authors.
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Supplementary Figures 19, Supplementary Table 1, Supplementary Notes 18, Supplementary Methods and Supplementary References. (PDF 8664 kb)
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Marshall, K., Jacobsen, C., Schäfermeier, C. et al. Continuousvariable quantum computing on encrypted data. Nat Commun 7, 13795 (2016). https://doi.org/10.1038/ncomms13795
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