Experimental realization of post-selected weak measurements on an NMR quantum processor
Dawei Lu, Aharon Brodutch, Jun Li, Hang Li, Raymond Laflamme
EExperimental realization of post-selected weak measurements on an NMRquantum processor
Dawei Lu, Aharon Brodutch, ∗ Jun Li,
1, 2
Hang Li,
1, 3 and Raymond Laflamme
1, 4, † Institute for Quantum Computing and Department of Physics & Astronomy,University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Department of Modern Physics, University of Scienceand Technology of China, Hefei, Anhui 230026, China Department of Physics, Tsinghua University, Beijing 100084, China Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: September 27, 2018)The ability to post-select the outcomes of an experiment is a useful theoretical concept andexperimental tool. In the context of weak measurements, post-selection can lead to surprisingresults such as complex weak values outside the range of eigenvalues. Usually post-selectionis realized by a projective measurement, which is hard to implement in ensemble systems suchas NMR. We demonstrate the first experiment of a weak measurement with post-selectionon an NMR quantum information processor. Our setup is used for measuring complex weakvalues and weak values outside the range of eigenvalues. The scheme for overcoming theproblem of post-selection in an ensemble quantum computer is general and can be applied toany circuit-based implementation. This experiment paves the way for studying and exploitingpost-selection and weak measurements in systems where projective measurements are hardto realize experimentally.
PACS numbers:
I. INTRODUCTION
Many fundamental experiments in quantum mechanics can be written and fully understood interms of a quantum circuit [1]. The current state of the art, however, provides a poor platform forimplanting these circuits on what can be considered a universal quantum computer. The limitednumber of independent degrees of freedom that can be manipulated efficiently limit the possibleexperiments. Many popular experiments involving a small number of qubits include some type of ∗ Electronic address: [email protected] † Electronic address: lafl[email protected] a r X i v : . [ qu a n t - ph ] J un post-selection. By post-selecting, experimenters condition their statistics only on those experimentsthat (will) meet a certain criteria such as the result of a projective measurement performed at theend of the experiment. Such conditioning produces a number of surprising effects like strangeweak values [2] and nonlinear quantum gates [3]. Although they are sometimes described viaquantum circuits, experiments involving post-selection are usually implemented in a dedicatedsetup which is not intended to act as a universal quantum processor. For example in currentoptical implementations [4, 5] the addition of more gates would require extra hardware.Nowadays, implementations of quantum computing tasks in NMR architectures provide a goodtest-bed for up to 12 qubits [6]. Universal quantum circuits can be implemented on mid-scale(5-7 qubit) NMR quantum computers with the development of high-fidelity control [7]. However,the difficulty in performing projective measurements poses a handicap with the implementation ofcircuits involving post-selection. Here we will show how to overcome this difficulty theoreticallyand experimentally, in the setting of weak measurements.Weak measurements provide an elegant way to learn something about a quantum system S inthe interval between preparation and post-selection [8]. When the interaction between S and ameasuring device M is weak enough, the back-action is negligible [9]. Moreover the effective evo-lution of the measuring device during the measurement is proportional to a weak value , a complexnumber which is a function of the pre-selection, post-selection and the desired observable. Theseweak values allow us to make statements that would otherwise be in the realm of counterfactuals [10, 11]. Weak values have been interpreted as complex probabilities [12] and/or element of reality [13] and, although somewhat controversial [14–16], they are used to describe a number of funda-mental issues in quantum mechanics including: Non-locality in the two slit experiment [9], thetrajectories of photons [17], the reality of the wave function[18], Hardy’s paradox[10, 11, 19], thethree box paradox [8, 13, 20], measurement precision-disturbance relations [5, 21] and the Leggett-Garg inequality [22–24]. Recently the amplification effect associated with large real and imaginaryweak values was used in practical schemes for precision measurements. While these schemes cannotovercome the limits imposed by quantum mechanics, they can be used to improve precision undervarious types of operational imitations such as technical noise [25].Experimental realizations of weak measurements have so far been limited to optics with only afew recent exceptions [24, 26]. In all cases post-selection was done using projective measurementsthat physically select events with successful post-selection. Optical experiments (e.g ref. [27])exploit the fact that one degree of freedom can be used as the system while another (usually acontinuous degree of freedom) can be used as the measuring device. Post-selection is achievedby filtering out photons that fail post-selection and the readout is done at the end only on thesurviving systems. This type of filtering is outside the scope of ensemble quantum computerswhere all operations apart from the final ensemble measurement are unitary. One can overcomethe difficulty by either including a physical filter or by finding some way for post-selection usingunitary operations. The former poses a technical challenge as well as a conceptual deviation fromthe circuit model. In what follows we show how to perform the latter by resetting the measuringdevice each time post-selection fails. Our theoretical proposal is compatible with a number ofcurrent implementation of quantum processors such as such as liquid and solid state NMR[6, 28–30], electron spin qubits [31, 32] and rare earth crystal implementations [33].We begin by describing the weak measurement process for qubits and the theoretical circuit forweak measurements on an ensemble quantum computer (Fig. 1). Next we describe our experimentin detail (Fig. 2). We demonstrate two properties of weak values associated with post-selection:weak values outside the range of eigenvalues and imaginary weak values (Figs. 3, 4). We concludewith a discussion of future applications II. WEAK MEASUREMENTS
The weak measurement procedure (for qubits) [34] involves a system S qubit initially prepared inthe state | ψ (cid:105) S and a measuring device qubit initially in the state |M i (cid:105) M such that (cid:104)M i | σ z |M i (cid:105) = 0.The system and measuring device are coupled for a very short time via the interaction Hamiltonian H i = δ ( t − t i ) gσ S ˆ n σ M z with the coupling constant g << σ ˆ n = ˆ n · (cid:126)σ a Pauli observable in thedirection ˆ n . This is followed by a projective (post-selection) measurement on S with outcome | φ (cid:105) S .Up to normalization we have | ψ, M i (cid:105) S , M → e − igσ S ˆ n σ M z | ψ, M i (cid:105) S , M → cos( g ) (cid:104) φ | ψ (cid:105) | φ, M i (cid:105) S , M − i sin( g ) (cid:104) φ | σ ˆ n | ψ (cid:105) σ M z | φ, M i (cid:105) S , M (1)The unnormalized state of M at the end is |M f (cid:105) = (cid:104) φ | ψ (cid:105) [cos( g ) M − i sin( g ) { σ ˆ n } w σ M z ] |M i (cid:105) S , M (2)where { σ ˆ n } w = (cid:104) φ | σ S ˆ n | ψ (cid:105)(cid:104) φ | ψ (cid:105) (3)is the weak value of σ ˆ n . We can now take the weak measurement approximation | g { σ ˆ n } w | << g we have |M f (cid:105) ≈ e − ig { σ ˆ n } w σ M z |M i (cid:105) (4)The measurement device is rotated by g { σ ˆ n } w around the z -axis. Note that this is the actualunitary evolution (at this approximation), not an average unitary. If we set the measuring deviceto be in the initial state √ [ | (cid:105) + | (cid:105) ], the expectation value for σ M ˆ m = ˆ m · (cid:126)σ M will be (cid:104)M f | σ M ˆ m |M f (cid:105) ≈ (cid:104)M i | ( ig { σ ˆ n } ∗ w σ M z ) σ M ˆ m ( − ig { σ ˆ n } w σ M z ) |M i (cid:105) (5) ≈ (cid:104)M i | σ M ˆ m |M i (cid:105) + ig (cid:104)M i | { σ ˆ n } ∗ w σ M z σ M ˆ m − { σ ˆ n } w σ M ˆ m σ M z |M i (cid:105) (6)= (cid:104)M i | σ M ˆ m |M i (cid:105) + igRe ( { σ ˆ n } w ) (cid:104)M i | σ M z σ M ˆ m − σ M ˆ m σ M z |M i (cid:105) + gIm ( { σ ˆ n } w ) (cid:104)M i | σ M z σ M ˆ m + σ M ˆ m σ M z |M i (cid:105) (7)Choosing ˆ m appropriately we can get the real or imaginary part of the weak value.The scheme above is based on the fact that S was post-selected in the correct state | φ (cid:105) . Sincethis cannot be guaranteed in an experiment, we need to somehow disregard those events wherepost-selection fails. As noted previously, the standard method for implementing the post-selectionis by filtering out the composite S , M systems that fail post-selection. While this is relativelysimple in some implementations it is not a generic property of a quantum processor. In particularpost-selection is not implicit in the circuit model. The method to overcome this problem is to resetthe measuring device whenever post-selection fails.One way to implement the controlled reset (see Fig. 1) is by using a controlled depolarizinggate with the following properties | φ (cid:105) (cid:104) φ | S ⊗ ρ M → | φ (cid:105) (cid:104) φ | S ⊗ ρ M | φ ⊥ (cid:105) (cid:104) φ ⊥ | S ⊗ ρ M → | φ ⊥ (cid:105) (cid:104) φ ⊥ | S ⊗ / , (8)where ρ M is an arbitrary state and | φ ⊥ (cid:105) S is the orthogonal state to the post-selection (cid:104) φ ⊥ | φ (cid:105) = 0.This gate is not unitary and requires an ancillary system A . It can be constructed by setting ρ A = / A and M . To keep the controlin the computational basis, we preceded the controlled-SWAP gate by U S† φ where U φ | (cid:105) = | φ (cid:105) .At the readout stage we need to measure the observable | (cid:105) (cid:104) | which will give the probabilityof post-selection p as well as (cid:104) σ M ˆ m (cid:105) . Finally we divide (cid:104) σ M ˆ m (cid:105) by p to obtain the weak value.If we are only interested in reading out the real or imaginary part of the weak value we canfurther simplify this operation by a controlled dephasing in a complementary basis to σ M ˆ m . Thisallows us to replace the controlled-SWAP by a controlled-controlled-phase with M as the targetand A , S as the control (see Fig. 2b). In our experimental setup we used the latter (simplified)approach. A controlled-controlled- σ z was used in the post-selection for measuring real weak valuesand controlled-controlled- σ x was used for complex weak values. DepolarizingPre-selection Post-selection ReadoutWeak measurementWeakMeasurement | Ψ > ρ i < σ z >< σ ̂ n > SystemMeasuring device U Φ† FIG. 1: (color online). Conceptual circuit for post-selected weak measurements. The system S is preparedin the state | ψ (cid:105) , it then interacts weakly with the measurement device M (the weak measurement stage).Post-selection is performed by a controlled depolarizing channel in the desired basis, Eq (8), decomposedinto a rotation U † φ followed by a contolled depolarizing in the computational basis. Finally at the readoutstage we get the probability of post-selection p from (cid:104) σ S z (cid:105) = 2 p −
1. The exception values (cid:104) σ M ˆ n (cid:105) are shiftedby a factor proportional to the weak value and decayed by a factor p . III. EXPERIMENTAL IMPLEMENTATION
We conducted three types of experiments. In each experiment we prepared S in an initial stateon the x − z plane, cos( θ ) | (cid:105) S + sin( θ ) | (cid:105) S , the weak measurement observable was a Pauli operatorin the x − y plane, cos( α ) σ S x +sin( α ) σ S y and the post-selection was always the | (cid:105) S state. In the firstexperiment (Fig. 3b) we set α = 0 to make a measurement of σ x and varied the coupling strengthfrom g = 0 .
05 to g = 0 .
7. We used three different initial states: θ = π/ g = π/
4) measurement, θ = 1 . { σ x } w = 2 .
57, at g = 0 .
05 , and θ = 1 .
4- where the weak value (first order) approximation is off by more than 10% at g = 0 .
05. Next wekept the coupling constant at g = 0 . θ to observe real weak values both insideand outside the range of eigenvalues [ − ,
1] (Fig. 4a). Finally we measured complex weak valueswith an absolute magnitude of 1 by keeping the initial state constant, θ = π/
4, and varying overthe measurement direction α at g = 0 . C labeled trichloroethylene (TCE) dissolved in d-chloroform. The struc-ture of the molecule is shown in Fig. 2a, where we denote C1 as qubit 1, C2 as qubit 2, and H asqubit 3. The internal Hamiltonian of this system can be described as H = (cid:88) j =1 πν j σ jz + π J σ z σ z + J σ z σ z )+ π J ( σ x σ x + σ y σ y + σ z σ z ) , (9)where ν j is the chemical shift of the j th spin and J ij is the scalar coupling strength between spins i and j . As the difference in frequencies between C1 and C2 is not large enough to adopt the (NMR[41]) weak J-coupling approximation [35], these two carbon spins are treated in the strongly coupledregime. The parameters of the Hamiltonian are obtained by iteratively fitting the calculated andobserved spectra through perturbation, and shown in the table of Fig. 2a.We label C1 as the ancilla A , C2 as the measuring device M , and H as the system S (Fig. 2b).Each experiment can be divided into four parts: (A) Pre-selection: Initializing the ancilla to theidentity matrix /
2, the measurement device to 1 / √ | (cid:105) + | (cid:105) ), and the system to cosθ | (cid:105) + sinθ | (cid:105) .(B) Weak measurement: Interaction between the measuring device and system, denoted by U w inthe network. (C) Post-selection of the system in the state | (cid:105) . (D) Measurement: (cid:104) σ M y (cid:105) on themeasuring device ( (cid:104) σ M z (cid:105) for the imaginary part) and (cid:104) σ S z (cid:105) on the system. The experimental detailsof the four parts above are as follows.(A) Pre-selection: Starting from the thermal equilibrium state, first we excite the ancilla C1 tothe transverse field by a π/ /
2. Next we create the pseudopure state (PPS) of C2 ( M ) andH ( S ) with deviation ⊗| (cid:105) (cid:104) | using the spatial average technique [30]. For traceless observablesand unital evolution the identity part of the PPS can be treated as noise on top of a pure state.The spectra of the PPS followed by π/ /
2, and these two peaksare utilized as the benchmark for the following experiments. Finally we apply one Hadamard gate C1 C2 H T (s) T (s)C1 21784.6 13.0 0.3 0.45 0.02C2 103.03 20528.0 8.9 0.3 1.18 0.02H 8.52 201.45 4546.9 8.9 0.3 1.7 0.2 H |0 〉 I U W R( θ ) Z M1M2 H R( θ ) U W Hadamardexp(-i θ σ y )exp(-i g σ x σ z ) Z exp(-i π /2 σ z )M1 σ y σ z M2 C1:C2:H: |0 〉 (a) (b) C1-Ancilla; C2-Measuring Device; H-System.
C2 C1HCl ClCl
FIG. 2: (color online). (a) Experimental implementation of a weak measurement in NMR using trichloroethy-lene in which the two C and one H spins form a 3-qubit sample. In the parameter table the diagonalelements are the chemical shifts (Hz), and the off-diagonal elements are scalar coupling strengths (Hz). T and T are the relaxation and dephasing time scales. (b) The quantum network used to realize the post-selected weak measurement in the experiment with C1 as the ancilla A , C2 as the measuring device M , andH as the system S . Post-selection of | (cid:105) S was achieved using the controlled-controlled-Z gate so that M is dephased when post-selection fails. The final measurements give the expectation values (cid:104) σ S z (cid:105) and (cid:104) σ M y (cid:105) which are used to calculate the weak value via Eq (13). on C2 and one R y ( θ ) = e − iθσ y rotation on H. At the end of this procedure the state is ρ ini = ⊗
12 ( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ) ⊗ (cos θ | (cid:105) + sin θ | (cid:105) )(cos θ (cid:104) | + sin θ (cid:104) | ) . (10)(B) Weak measurement: The unitary operator to realize the weak measurement U w = e − igσ n σ z . (11)can be simulated by the interaction term σ z σ z between C2 and H using the average Hamiltoniantheory [36]. However, since the internal Hamiltonian contains a strongly coupling term and therefocusing scheme requires the WAHUHA-4 sequence [37] we adopted the gradient ascent pulseengineering (GRAPE) technique [38, 39] to improve the fidelity (see discussion below).(C) Post-selection: In order to mimic the post-selection of | (cid:105) on the system spin H in NMR,we introduce an ancilla qubit C1 in the maximally mixed state
1. The controlled resetting noiseoperation is a controlled-controlled- σ z gate ⊗ ⊗ − | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ⊗ σ z . (12)If post-selection is successful, the measurement device will point to the weak value. Otherwisethe measurement device will become dephased and the expectation value (cid:10) σ y (cid:11) will be reset to 0.For the measurement of complex weak values we used a standard Toffoli and applied the samereasoning.(D) Measurement: Finally we measure the expectation value (cid:104) σ y (cid:105) on C2 and (cid:104) σ z (cid:105) on H, tocalculate the weak value by the expression Re ( { σ x } w ) ≈ (cid:104) σ y (cid:105) g ( (cid:104) σ z (cid:105) + 1) . (13)For the imaginary part we similarly use (cid:10) σ z (cid:11) .Since the timescales for the experiment are much shorter than T the evolution is very close tounital.In the experiment, the π/ µ s.All the other operations are implemented through GRAPE pulses to achieve high-fidelity control.These GRAPE pulses are designed to be robust to the inhomogeneity of the magnetic field, andthe imprecisions of the parameters in the Hamiltonian. For the two selective π/ θ and interaction strength g in the experiment, we have used different GRAPE pulses to implementthe experiments with different group of parameters. All of these GRAPE pulses have the samelength: 20 ms, and fidelity over 99.98%.There are two essential advantages in utilizing the GRAPE pulses: reducing the error accu-mulated by the long pulse sequence if we directly decompose the network, and reducing the errorcaused by decoherence. Because the U w evolution is supposed to be very “weak”, the intensity ofthe output signal is quite small. Moreover, the calculated weak values are more sensitive to theintensity of the signal, because the small g in the denominator (Eq 13) will amplify the errors inthe experiment. Therefore, we need to achieve as accurate coherent control as we can to obtainprecise experimental results. The direct decomposition of the original network requires many singlerotations, as well as a relatively long evolution time. For example, an efficient way to decomposethe controlled-controlled- σ z gate is by six CNOT gates and several one qubit gates [40], whichrequires more than 50 ms in our experimental condition. Therefore, using short GRAPE pulses of high fidelity we were able to decrease the potential errors due to decoherence and imperfect imple-mentation of the long pulse sequence. Fig. 3a shows the experimental spectra compared with thesimulated one in the case g = 0 .
05 and θ = 1 .
4. The top plots are the reference (PPS) spectra whilethe bottom are the spectra measured at the end of the experiment. The predicted value of (cid:104) σ y (cid:105) inthis case is only 1.67%. However, since the experimental spectra are very close to the simulatedone, we can obtain a good result after extracting the data. The effects of decoherence in the caseof small post-selection probabilities values can be seen in Fig. 3b where the experimental valuesare consistently lower than the theoretical predictions. Since decoherence reduces the expectationvalues the results of Eq (13) will be more sensitive to decoherence as (cid:104) σ z (cid:105) → − g = 0 .
05 to g = 0 . α = 0 andthree different initial states θ = 1 . θ = 1 .
2, and θ = π/
4. We obtained the weak value througha final measurement of (cid:104) σ y (cid:105) on the measuring device C2 and (cid:104) σ z (cid:105) on the system H, respectively.The weak value { σ x } w was calculated through Eq (13), with the result shown in Fig. 3b. Theerror bars were obtained by repeating the experiment four times. The large error bars in the weakvalues at small values of g and low post-selection probabilities are a result of imperfect calibrationof the low experimental signals obtained at that range.In the second experiment we studied the behavior of measured weak values as a function of theinitial state parameter θ at g = 0 . α = 0. The theoretical weak value should be tan( θ ) (at g → g = 0 . θ ) as the overlap between the pre and post-selectionvanishes and second order terms become more dominant. The experimental data matches with thesmooth part of the curve. When θ is very close to π/
2, we were unable to measure the extremelylow NMR signals due to the signal-to-noise ratio(SNR) issues, moreover decoherence effects becomemore prominent at these values (see discussion above).In the final experiment we measured both the real part and imaginary parts of the weak value(Fig. 4b). We set θ = π/ g = 0 .
1, and changed α , the observable of the measuring device C2.The expected weak values have an absolute magnitude of 1 and the overlap between the pre andpost-selected state is 1 / √
2. To measure the imaginary part, we replaced the controlled-controlled- σ z gate in Fig. 2b with a controlled-controlled- σ x gate, and measured the expectation value (cid:104) σ z (cid:105) on the measuring device C2. The other parts of the experiments remain the same. In Fig. 2bwe can see the real part and imaginary part of the weak value along with the parameter α . The0 (a) (b) g w e a k v a l u e < σ x > Initial State cos θ |0>+sin θ |1> θ = 1.4 (Theo) θ = 1.4 (Exp) θ = 1.2 (Theo) θ = 1.2 (Exp) θ = π /4 (Theo) θ = π /4 (Exp) NMR Frequency (Hz) N M R s i g n a l ( A r b . U n i t ) C2 Spectra simulationexperiment
NMR Frequency (Hz) N M R s i g n a l ( A r b . U n i t ) H Spectra simulationexperiment
PPS PPSg=0.05; θ =1.4 g=0.05; θ =1.4 FIG. 3: (color online). (a) The spectra for each qubit (bottom spectrum) was compared with the referencePPS (top spectrum) to obtain the expectation values (cid:104) σ M y (cid:105) on C2 and (cid:104) σ S z (cid:105) on H. To observe the signal on H,we applied π/ g = 0 .
05 and θ = 1 .
4, thesimulated spectra (blue) fit well with the experimental one (red). (b) Weak values for various initial states,cos( θ ) | (cid:105) S +sin( θ ) | (cid:105) S , were calculated using Eq (13) and compared the theoretical predictions as a functionof the measurement strength from g = 0 .
05 to g = 0 .
7. The solid curves are theoretical predictions withoutthe weak measurement approximation. When the overlap between the pre and post-selection, cos( θ ) is largeenough we get very close to the asymptotic g → g ≥ .
05. However for θ = 1 . g = 0 .
05. The error bars are plotted by repeating each experiment four times. Atlow post-selection probabilities we see observed values decrease due to decoherence. α (g=0.1) w e a k v a l u e < c o s α σ x - s i n α σ y > Initial
State cos( π /4)|0>+sin( π /4)|1> RealImag θ (g=0.1) w e a k v a l u e < σ x > Initial
State cos θ |0>+sin θ |1> simulation experiment (a) (b) FIG. 4: (color online). (a) Experimental weak values { σ x } w as a function of the initial state cos( θ ) | (cid:105) +sin( θ ) | (cid:105) for a fixed post-selection | (cid:105) at g = 0 .
1. The theoretical (blue) curve is plotted for g = 0 . π/ < θ < π − π/ − ≤ (cid:104) σ x (cid:105) ≤ θ approached π/ { cos( α ) σ x − sin( α ) σ y } w at g = 0 . θ = π/ α . At these values it was possible to get the full range of weakvalues with an absolute magnitude of 1. results match well with the theory. IV. CONCLUSIONS AND OUTLOOK
Post-selection is a useful and interesting conceptual and practical tool. Its experimental im-plementations have so far been limited to dedicated quantum devices. Here we showed that thisparadigm can be realized in a more general setting of a quantum circuit with unitary gates. Weimplemented a weak measurement where shifts corresponding to weak values outside the range ofeigenvalues ([ − ,
1] in the case of Pauli observables) are an artifact of non trivial post-selection.Our experiment involved a simple 3-qubit system and we were able to demonstrate the mea-surement of large (Fig 3b, 4a) and imaginary (Fig. 4b) weak values. We observed a shift of 2 . g with an accuracy of ± . g at θ = 1 . g = 0 .